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Graphs/122217: If f (x) = 5x – 1, find the following: 9. f (a -2) 1 solutions Answer 89710 by jim_thompson5910(28715)   on 2008-01-23 23:28:11 (Show Source): You can put this solution on YOUR website! Let's evaluate Start with the given function. Plug in . In other words, replace each x with . Distribute Combine like terms

Graphs/122216: If f (x) = 4x -3, find the following: 8. f (-1) 1 solutions Answer 89709 by jim_thompson5910(28715)   on 2008-01-23 23:25:26 (Show Source): You can put this solution on YOUR website! Let's evaluate Start with the given function. Plug in . In other words, replace each x with -1. Multiply 4 and -1 to get -4 Now combine like terms

Graphs/122215: Graph the function for each. 7. f (x) = -2x – 5 1 solutions Answer 89708 by jim_thompson5910(28715)   on 2008-01-23 23:23:59 (Show Source): You can put this solution on YOUR website! Looking at we can see that the equation is in slope-intercept form where the slope is and the y-intercept is Since this tells us that the y-intercept is .Remember the y-intercept is the point where the graph intersects with the y-axis So we have one point Now since the slope is comprised of the "rise" over the "run" this means Also, because the slope is , this means: which shows us that the rise is -2 and the run is 1. This means that to go from point to point, we can go down 2 and over 1 So starting at , go down 2 units and to the right 1 unit to get to the next point Now draw a line through these points to graph So this is the graph of through the points and

Graphs/122214: Rewrite each equation as a function of x. 6. -3x + 4y = 11 1 solutions Answer 89707 by jim_thompson5910(28715)   on 2008-01-23 23:22:05 (Show Source): You can put this solution on YOUR website! Solved by pluggable solver: Converting Linear Equations in Standard form to Slope-Intercept Form (and vice versa) Start with the given equation Add to both sides Multiply both sides by Distribute Multiply Rearrange the terms Reduce any fractions So the equation is now in slope-intercept form () where (the slope) and (the y-intercept)

Graphs/122213: Evaluate each function for the value specified. 5. f (x) = x2 - 7x + 10; find (a) f (0), (b) f (5), and (c) f (-2). 1 solutions Answer 89706 by jim_thompson5910(28715)   on 2008-01-23 23:20:12 (Show Source): You can put this solution on YOUR website! a) Let's evaluate Start with the given function. Plug in . In other words, replace each x with 0. Evaluate to get 0. Multiply -7 and 0 to get 0 Now combine like terms b) Let's evaluate Start with the given function. Plug in . In other words, replace each x with 5. Evaluate to get 25. Multiply -7 and 5 to get -35 Now combine like terms c) Let's evaluate Start with the given function. Plug in . In other words, replace each x with -2. Evaluate to get 4. Multiply -7 and -2 to get 14 Now combine like terms

Graphs/122212: 4. Number problems. You have at least $30 in change in your drawer, consisting of dimes and quarters. Write an inequality that shows the different number of coins in your drawer. 1 solutions Answer 89705 by jim_thompson5910(28715)   on 2008-01-23 23:17:35 (Show Source): You can put this solution on YOUR website! Let d=# of dimes, q=# of quarters, s=sum of your money Since a dime is 10 cents, it is $0.10 by conversion from cents to dollars. Also a quarter, which is 25 cents, is $0.25 So the sum of your money would be Since you have at least $30, this means that the sum of your money is greater than $30 like this: Now plug in Answer: So the inequality that represents this problem is

Graphs/122210: Graph each of the following inequalities. 3. 3x – 4y > 12 1 solutions Answer 89703 by jim_thompson5910(28715)   on 2008-01-23 23:09:13 (Show Source): You can put this solution on YOUR website! In order to graph , we need to graph the equation (just replace the inequality sign with an equal sign). So lets graph the line (note: if you need help with graphing, check out this solver) graph of Now lets pick a test point, say (0,0). Any point will work, (just make sure the point doesn't lie on the line) but this point is the easiest to work with. Now evaluate the inequality with the test point Substitute (0,0) into the inequality Plug in and Simplify (note: for some reason, some of the following images do not display correctly in Internet Explorer. So I recommend the use of Firefox to see these images.) Since this inequality is not true, we do not shade the entire region that contains (0,0). So this means we shade the region that is on the opposite side of the line Graph of with the boundary (which is the line in red) and the shaded region (in green) (note: since the inequality contains a greater-than sign, this means the boundary is excluded. This means the solid red line is really a dashed line)

Graphs/122209: Graph each of the following inequalities. 2. 4x + y ≥ 4 1 solutions Answer 89702 by jim_thompson5910(28715)   on 2008-01-23 23:06:45 (Show Source): You can put this solution on YOUR website! In order to graph , we need to graph the equation (just replace the inequality sign with an equal sign). So lets graph the line (note: if you need help with graphing, check out this solver) graph of Now lets pick a test point, say (0,0). Any point will work, (just make sure the point doesn't lie on the line) but this point is the easiest to work with. Now evaluate the inequality with the test point Substitute (0,0) into the inequality Plug in and Simplify (note: for some reason, some of the following images do not display correctly in Internet Explorer. So I recommend the use of Firefox to see these images.) Since this inequality is not true, we do not shade the entire region that contains (0,0). So this means we shade the region that is on the opposite side of the line Graph of with the boundary (which is the line in red) and the shaded region (in green)

Graphs/122208: We have graphed the boundary line for the linear inequality. Determine the correct half – plane in each case, and complete the graph. y>3 1 solutions Answer 89701 by jim_thompson5910(28715)   on 2008-01-23 23:03:41 (Show Source): You can put this solution on YOUR website! Start with the graph of graph of Now lets pick a test point, say (0,0). Any point will work, (just make sure the point doesn't lie on the line) but this point is the easiest to work with. Now evaluate the inequality with the test point Substitute (0,0) into the inequality Plug in and Simplify (note: for some reason, some of the following images do not display correctly in Internet Explorer. So I recommend the use of Firefox to see these images.) Since this inequality is not true, we do not shade the entire region that contains (0,0). So this means we shade the region that is on the opposite side of the line Graph of with the boundary (which is the line in red) and the shaded region (in green) (note: since the inequality contains a greater-than sign, this means the boundary is excluded. This means the solid red line is really a dashed line) So the correct half-plane is above the line

Graphs/122153: I am stuck on this linear equation. I also need to find the rise over run too. This is what I have so far. 2x + 4y + 1 = 0 4y = 2x - 1 /4 /4 /4 y = 2/4x -1/4 y = 1/2x -1/4 1 solutions Answer 89673 by jim_thompson5910(28715)   on 2008-01-23 19:48:28 (Show Source): You can put this solution on YOUR website! You made a mistake in converting to slope-intercept form "4y = 2x - 1" <--- In this step, it should be -2x since you subtract 2x from both sides So the equation in slope-intercept form is Looking at we can see that the equation is in slope-intercept form where the slope is and the y-intercept is Since this tells us that the y-intercept is .Remember the y-intercept is the point where the graph intersects with the y-axis So we have one point Now since the slope is comprised of the "rise" over the "run" this means Also, because the slope is , this means: which shows us that the rise is -1 and the run is 2. This means that to go from point to point, we can go down 1 and over 2 So starting at , go down 1 unit and to the right 2 units to get to the next point Now draw a line through these points to graph So this is the graph of through the points and

absolute-value/122134: I don't know the absolute value of -10. So can you please help me and also explain it to me. THANK YOU!!!!! 1 solutions Answer 89671 by jim_thompson5910(28715)   on 2008-01-23 19:39:33 (Show Source): You can put this solution on YOUR website! Since the distance from -10 to 0 is 10 units, this means Remember, absolute value refers to distance. So it only makes sense to have positive answers.

Expressions-with-variables/122150: I think this is the right section. My father in law is trying to help someone with their homework and called me. My algebra memory has faded :) solve for x x=zb-zk^2 (or, in english, x equals zb minus zk squared) 1 solutions Answer 89669 by jim_thompson5910(28715)   on 2008-01-23 19:30:54 (Show Source): You can put this solution on YOUR website! Since x is alone on the left side it is already isolated and solved for. So no work needs to be done if you want to solve for x. Do you want to solve for another variable?

Graphs/122142: solve the following systems by using either addition or substitution. if a unique solution does not exist, state whether the system is dependent of inconsistent. 10x+2y=7 y=-5x+3 1 solutions Answer 89666 by jim_thompson5910(28715)   on 2008-01-23 19:13:05 (Show Source): You can put this solution on YOUR website! Start with the given system Plug in into the first equation. In other words, replace each with . Notice we've eliminated the variables. So we now have a simple equation with one unknown. Distribute Combine like terms on the left side Subtract 6 from both sides Combine like terms on the right side Since this equation is never true for any x value, this means there are no solutions. So the system is inconsistent.

Number-Line/122135: The problem is this: -5 is greater than/equal to x-6 1 solutions Answer 89661 by jim_thompson5910(28715)   on 2008-01-23 18:53:01 (Show Source): You can put this solution on YOUR website! Start with the given inequality Add 5 to both sides Subtract x from both sides Combine like terms on the right side Divide both sides by -1 to isolate x (note: Remember, dividing both sides by a negative number flips the inequality sign) Divide -------------------------------------------------------------- Answer: So our answer is

Trigonometry-basics/122078: #10 Find the coordinates [r, theta] of all points of intersection: r = 2sin(theta) and r = 1 a) [½, pi/3] and [½, 5pi/3] b) [½, pi/6] and [½, 5pi/6] c) [1, pi/3] and [1, 5pi/3] d) [1, pi/6] and [1, 5pi/6] e) None of these 1 solutions Answer 89607 by jim_thompson5910(28715)   on 2008-01-23 14:29:00 (Show Source): You can put this solution on YOUR website! Start with the given set of equations Plug in into the first equation Divide both sides by 2 Take the arcsine of both sides to isolate or Take the arcsine of to get and : So because the answer format is this means we have the solutions: and So the answer is D) Note: Since the second equation is , this means that the first coordinate (which is r) is also 1. So the answers will look like (1,?) and (1,?). So if you had no idea what to do, you could easily eliminate possible answers a) and b)

Quadratic-relations-and-conic-sections/122072: #9 Identify x = 4sint y = 5cost a) circle b) parabola c) ellipse d) hyperbola e) none of these 1 solutions Answer 89606 by jim_thompson5910(28715)   on 2008-01-23 14:10:01 (Show Source): You can put this solution on YOUR website! Start with the first parametric equation Divide both sides by 4 to isolate . Start with the second parametric equation Divide both sides by 5 to isolate . Now we're going to use the trig identity: Replace with . Replace with . This is why we isolated sine and cosine. Square to get . Square to get . Notice how the equation is now in the form (note: h and k are equal to zero in this case) So this shows us that graphs an ellipse (since the above equation is the general equation of an ellipse). So this means that the two parametric equations also graph an ellipse. So the answer is C)

Complex_Numbers/122065: Find (1 / sq. rt. 2 + i / sq. rt. 2) ^ 8 a.) -8 b.) 256 c.) 8 d.) 1 e.) None of these 1 solutions Answer 89603 by jim_thompson5910(28715)   on 2008-01-23 13:50:16 (Show Source): You can put this solution on YOUR website! Start with the given expression It's very useful to note that and . So the expression is equivalent to Now we're going to use De Moivre's theorem to solve this problem. Remember, De Moivre's theorem states: So using De Moivre's theorem we get Multiply and to get . Now reduce to get Take the cosine of to get 1. Take the sine of to get 0. Remove the zero term So simplifies to 1. In other words, So the answer is D)

Linear-equations/122018: Find the equation of the line through the two points (5,3) and (-7,7). 1 solutions Answer 89574 by jim_thompson5910(28715)   on 2008-01-23 10:34:54 (Show Source): You can put this solution on YOUR website! First lets find the slope through the points (,) and (,) Start with the slope formula (note: is the first point (,) and is the second point (,)) Plug in ,,, (these are the coordinates of given points) Subtract the terms in the numerator to get . Subtract the terms in the denominator to get Reduce So the slope is ------------------------------------------------ Now let's use the point-slope formula to find the equation of the line: ------Point-Slope Formula------ where is the slope, and is one of the given points So lets use the Point-Slope Formula to find the equation of the line Plug in , , and (these values are given) Distribute Multiply and to get Add to both sides to isolate y Combine like terms and to get (note: if you need help with combining fractions, check out this solver) ------------------------------------------------------------------------------------------------------------ Answer: So the equation of the line which goes through the points (,) and (,) is: The equation is now in form (which is slope-intercept form) where the slope is and the y-intercept is Notice if we graph the equation and plot the points (,) and (,), we get this: (note: if you need help with graphing, check out this solver) Graph of through the points (,) and (,) Notice how the two points lie on the line. This graphically verifies our answer.

Pythagorean-theorem/121959: 1. A customer wants to make a teepee in his backyard for his children. He plans to use lengths of PVC plumbing pipe for the supports on the teepee, and he wants the teepee to be 12 feet across and 8 feet tall. How long should the PVC plumbing pipe be? 1 solutions Answer 89539 by jim_thompson5910(28715)   on 2008-01-23 00:32:37 (Show Source): You can put this solution on YOUR website! If we cut the triangle in half vertically down the middle, we get this triangle: Since we can see that the triangle has legs of 8 and 6 with a hypotenuse of x, we can use Pythagoreans theorem to find the unknown side. Pythagoreans theorem: where a and b are the legs of the triangle and c is the hypotenuse Plug in a=8, b=6, and c=x. Now lets solve for x Square each individual term Combine like terms Take the square root of both sides Simplify the square root So the hypotenuse of this triangle is 10 Since we cut the triangle in half, this means the other triangle's hypotenuse also has a length of 10 After putting the two triangles back together, we have this triangle: Since the figure was pointing to one side, I'm assuming that the problem is asking about one side. So the length of one pipe is 10 ft.

Trigonometry-basics/121956: 7) For x E [0,2pi), solve: cos2x = cosx a) 2 pi/3, 4pi/3 and pi b) 2 pi/3, 4 pi/3 and 0 c) Pi/3, 5pi/3 and 0 d) Pi/6, 5pi/6 and pi e) 5pi/6, 7pi/6 and pi 1 solutions Answer 89537 by jim_thompson5910(28715)   on 2008-01-23 00:08:41 (Show Source): You can put this solution on YOUR website! Start with the given equation Replace with . Note: I'm using the identity Subtract from both sides Rearrange the terms Now let Replace each with Factor the left side Now set each factor equal to zero or Now solve for u in each case or Now remember we let . So this means or So let's solve to get or . However, since is excluded the only solution for is Now let's solve to get or So putting these solutions together, we get: , or So this means the answer is B)

Trigonometry-basics/121955: 6) find csc(arctan 8/15) a) 17/8 b) 8/17 c) 15/17 d) 17/15 e) Undefined 1 solutions Answer 89535 by jim_thompson5910(28715)   on 2008-01-23 00:01:59 (Show Source): You can put this solution on YOUR website! First a triangle with legs of 8 and 15. Let x be the angle we'll refer to. By using pythagorean's theorem, we find that the hypotenuse is 17 units Since this means (notice how the arctangent is gives you an angle) Now remember and (remember the cosecant and sine function are reciprocals of each other) So which also means So the answer is A)

Trigonometry-basics/121954: 5) Find the period of f(x) = 3 - 2cos(3x+pi) a) 2 pi / 3 b) Pi / 3 c) Pi / 6 d) Pi e) 2 pi 1 solutions Answer 89533 by jim_thompson5910(28715)   on 2008-01-22 23:55:31 (Show Source): You can put this solution on YOUR website! The general form of cosine is: where the period So for the trig function which also looks like . Notice how matches up with , so which means Now simply plug in into the period formula So the period is So the answer is A)

Quadratic-relations-and-conic-sections/121913: Name the vertex for the parabol with equation y= 3x^2-12x+16 1 solutions Answer 89531 by jim_thompson5910(28715)   on 2008-01-22 23:50:55 (Show Source): You can put this solution on YOUR website! To find the vertex, we first need to find the axis of symmetry (ie the x-coordinate of the vertex) To find the axis of symmetry, use this formula: From the equation we can see that a=3 and b=-12 Plug in b=-12 and a=3 Negate -12 to get 12 Multiply 2 and 3 to get 6 Reduce So the axis of symmetry is So the x-coordinate of the vertex is . Lets plug this into the equation to find the y-coordinate of the vertex. Start with the given polynomial Plug in Raise 2 to the second power to get 4 Multiply 3 by 4 to get 12 Multiply 12 by 2 to get 24 Now combine like terms So the vertex is (2,4)

Polynomials-and-rational-expressions/121920: Solve for a. 6a(a+3) = 2a(3a+5)+16 1 solutions Answer 89530 by jim_thompson5910(28715)   on 2008-01-22 23:47:43 (Show Source): You can put this solution on YOUR website! Start with the given equation Distribute Get all of the terms to the left side Combine like terms Add 16 to both sides Combine like terms on the right side Divide both sides by 8 to isolate a Divide -------------------------------------------------------------- Answer: So our answer is

Linear-equations/121921: I need help finding slope of line passing through (-3,3) and (-3,-3) 1 solutions Answer 89528 by jim_thompson5910(28715)   on 2008-01-22 23:45:05 (Show Source): You can put this solution on YOUR website! Solved by pluggable solver: Finding the slope Slope of the line through the points (-3, 3) and (-3, -3) Since you CANNOT divide by 0, this means that the slope of the line is undefined. Answer: The slope of the line is undefined.

Graphs/121930: solve the following systems by addition. If a unique solution does not exist, state whether the system is inconsistent or dependent. 2x+3y=1 5x+3y=16 1 solutions Answer 89525 by jim_thompson5910(28715)   on 2008-01-22 23:44:08 (Show Source): You can put this solution on YOUR website! Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition Lets start with the given system of linear equations In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for y, we would have to eliminate x (or vice versa). So lets eliminate x. In order to do that, we need to have both x coefficients that are equal but have opposite signs (for instance 2 and -2 are equal but have opposite signs). This way they will add to zero. So to make the x coefficients equal but opposite, we need to multiply both x coefficients by some number to get them to an equal number. So if we wanted to get 2 and 5 to some equal number, we could try to get them to the LCM. Since the LCM of 2 and 5 is 10, we need to multiply both sides of the top equation by 5 and multiply both sides of the bottom equation by -2 like this: Multiply the top equation (both sides) by 5 Multiply the bottom equation (both sides) by -2 So after multiplying we get this: Notice how 10 and -10 add to zero (ie ) Now add the equations together. In order to add 2 equations, group like terms and combine them Notice the x coefficients add to zero and cancel out. This means we've eliminated x altogether. So after adding and canceling out the x terms we're left with: Divide both sides by to solve for y Reduce Now plug this answer into the top equation to solve for x Plug in Multiply Subtract from both sides Combine the terms on the right side Multiply both sides by . This will cancel out on the left side. Multiply the terms on the right side So our answer is , which also looks like (, ) Notice if we graph the equations (if you need help with graphing, check out this solver) we get graph of (red) (green) (hint: you may have to solve for y to graph these) and the intersection of the lines (blue circle). and we can see that the two equations intersect at (,). This verifies our answer.

Pythagorean-theorem/121931: A 15 feet ladder is 5 feet away from a wall. How far above the floor is the top of the ladder? 1 solutions Answer 89524 by jim_thompson5910(28715)   on 2008-01-22 23:43:20 (Show Source): You can put this solution on YOUR website! We basically have this triangle set up: Since we can see that the triangle has legs of x and 5 with a hypotenuse of 15, we can use Pythagoreans theorem to find the unknown side. Pythagoreans theorem: where a and b are the legs of the triangle and c is the hypotenuse Plug in a=x, b=5, and c=15. Now lets solve for x Square each individual term Subtract 25 from both sides Combine like terms Take the square root of both sides Simplify the square root Which approximates to... So our answer is So the top of the ladder is about 14.14 feet about the floor

Quadratic_Equations/121932: solve the following systems by addition. If a unique solution does not exist, state whether the system is inconsistent or dependent. X+5Y=10 -2X-10Y=-20 1 solutions Answer 89523 by jim_thompson5910(28715)   on 2008-01-22 23:41:53 (Show Source): You can put this solution on YOUR website! Solved by pluggable solver: Solving a linear system of equations by subsitution Lets start with the given system of linear equations Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to choose y. Solve for y for the first equation Subtract from both sides Divide both sides by 5. Which breaks down and reduces to Now we've fully isolated y Since y equals we can substitute the expression into y of the 2nd equation. This will eliminate y so we can solve for x. Replace y with . Since this eliminates y, we can now solve for x. Distribute -10 to Multiply Reduce any fractions Add to both sides Combine the terms on the right side Now combine the terms on the left side. Since this expression is true for any x, we have an identity. So there are an infinite number solutions. The simple reason is the 2 equations represent 2 lines that overlap each other. So they intersect each other at an infinite number of points. If we graph and we get graph of graph of (hint: you may have to solve for y to graph these) we can see that these two lines are the same. So this system is dependent

Linear-systems/121939: I need help with the following problems: y = 2x + 6 y = -x -3 y - 3x = 9 2y + x = 4 y + 4 = 2x 6x - 3y = 12 1 solutions Answer 89522 by jim_thompson5910(28715)   on 2008-01-22 23:40:57 (Show Source): You can put this solution on YOUR website! #1 Start with the given system Plug in into the first equation. In other words, replace each with . Notice we've eliminated the variables. So we now have a simple equation with one unknown. Distribute Combine like terms on the left side Add 3 to both sides Subtract 2x from both sides Combine like terms on the left side Combine like terms on the right side Divide both sides by -3 to isolate x Divide Now that we know that , we can plug this into to find Substitute for each Simplify So our answer is and which also looks like Notice if we graph the two equations, we can see that their intersection is at . So this verifies our answer. Graph of (red) and (green) #2 Solved by pluggable solver: Solving a linear system of equations by subsitution Lets start with the given system of linear equations Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to choose y. Solve for y for the first equation Add to both sides Divide both sides by 1. Which breaks down and reduces to Now we've fully isolated y Since y equals we can substitute the expression into y of the 2nd equation. This will eliminate y so we can solve for x. Replace y with . Since this eliminates y, we can now solve for x. Distribute 2 to Multiply Reduce any fractions Subtract from both sides Combine the terms on the right side Now combine the terms on the left side. Multiply both sides by . This will cancel out and isolate x So when we multiply and (and simplify) we get <---------------------------------One answer Now that we know that , lets substitute that in for x to solve for y Plug in into the 2nd equation Multiply Add to both sides Combine the terms on the right side Multiply both sides by . This will cancel out 2 on the left side. Multiply the terms on the right side Reduce So this is the other answer <---------------------------------Other answer So our solution is and which can also look like (,) Notice if we graph the equations (if you need help with graphing, check out this solver) we get graph of (red) and (green) (hint: you may have to solve for y to graph these) intersecting at the blue circle. and we can see that the two equations intersect at (,). This verifies our answer. ----------------------------------------------------------------------------------------------- Check: Plug in (,) into the system of equations Let and . Now plug those values into the equation Plug in and Multiply Add Reduce. Since this equation is true the solution works. So the solution (,) satisfies Let and . Now plug those values into the equation Plug in and Multiply Add Reduce. Since this equation is true the solution works. So the solution (,) satisfies Since the solution (,) satisfies the system of equations this verifies our answer. #3 Start with the first equation Solve for y by subtracting 4 from both sides Start with the given system Plug in into the first equation. In other words, replace each with . Notice we've eliminated the variables. So we now have a simple equation with one unknown. Distribute Combine like terms on the left side Subtract 12 from both sides Combine like terms on the right side Simplify Since this equation is always true for any x value, this means x can equal any number. So there are an infinite number of solutions.

Graphs/121953: This question is from textbook x+5/2=6 1 solutions Answer 89518 by jim_thompson5910(28715)   on 2008-01-22 23:35:47 (Show Source): You can put this solution on YOUR website! Start with the given equation Multiply both sides by the LCM of 2. This will eliminate the fractions (note: if you need help with finding the LCM, check out this solver) Distribute and multiply the LCM to each side Subtract 5 from both sides Combine like terms on the right side Divide both sides by 2 to isolate x -------------------------------------------------------------- Answer: So our answer is (which is approximately in decimal form)

Quadratic_Equations/121936: SOLVE BY SUBSTITUTION 5X-2Y=-5 Y-5X=3 1 solutions Answer 89516 by jim_thompson5910(28715)   on 2008-01-22 23:34:55 (Show Source): You can put this solution on YOUR website! Solved by pluggable solver: Solving a linear system of equations by subsitution Lets start with the given system of linear equations Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to choose y. Solve for y for the first equation Subtract from both sides Divide both sides by -2. Which breaks down and reduces to Now we've fully isolated y Since y equals we can substitute the expression into y of the 2nd equation. This will eliminate y so we can solve for x. Replace y with . Since this eliminates y, we can now solve for x. Distribute 1 to Multiply Reduce any fractions Subtract from both sides Make 3 into a fraction with a denominator of 2 Combine the terms on the right side Make -5 into a fraction with a denominator of 2 Now combine the terms on the left side. Multiply both sides by . This will cancel out and isolate x So when we multiply and (and simplify) we get <---------------------------------One answer Now that we know that , lets substitute that in for x to solve for y Plug in into the 2nd equation Multiply Subtract from both sides Combine the terms on the right side Multiply both sides by . This will cancel out 1 on the left side. Multiply the terms on the right side Reduce So this is the other answer <---------------------------------Other answer So our solution is and which can also look like (,) Notice if we graph the equations (if you need help with graphing, check out this solver) we get graph of (red) and (green) (hint: you may have to solve for y to graph these) intersecting at the blue circle. and we can see that the two equations intersect at (,). This verifies our answer. ----------------------------------------------------------------------------------------------- Check: Plug in (,) into the system of equations Let and . Now plug those values into the equation Plug in and Multiply Add Reduce. Since this equation is true the solution works. So the solution (,) satisfies Let and . Now plug those values into the equation Plug in and Multiply Add Reduce. Since this equation is true the solution works. So the solution (,) satisfies Since the solution (,) satisfies the system of equations this verifies our answer.