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Are you sure that you wrote dpwn the problem correctly?

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Since x+1 = y AND y=x^2-4x+7, x+1 = x^2-4x+7
This equation can be rewritten as 0 = x^2-5x+6.
This equation can be factored into linear expressions so x^2-5x+6=(x-3)*(x-2)=0
So x=3 and x = 2.
Next each of values for x can be substituted in either equation to determine the values of y.
Using the equation y = x +1.
When x = 2, y = 2+1 = 3
When x = 3, y = 3+1 = 4.
So, there are two solutions to the system of equations.
You may want to graph both equation to "see" the points of intersection of the two graphs.

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Since you making "4 times" the recipe, you need to multiply 3 2/3 by 4.
First convert the mixed number 3 2/3 to an improper fraction.
3 2/3 = 11/3
So 3 2/3 * 4 = (11/3) * 4 = 44/3 (Tablespoons).
Now, convert the improper fraction 44/3 to a mixed number. 44 divided by 3 = 14 and remainder of 2. So 44/3 = 14 2/3 (Tablespoons)

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This question tests your ability to understand the equivalence of number represented as percents, decimals and fractions.
First, convert 60% to a decimal number. To do this, drop the "%" sign and divide by 100. So 60% = .60
Next, either convert each of the fractions that are the possible answer to decimals, for example 3/50 = .06
or convert .60 to a fraction in lowest terms. .60 = 60/100 = 3/5
So, 60% = .60 = 60/100 = 3/5

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The x-intercepts of the equation can be found by finding the roots of the equation x^2+4x-1. Use the quadratic formula x =( -b +/- Square Root[b^2-4ac])/(2*a)
where a= 1, b=4 and c=-1.
Sunstituting these values intp the quadratic formula we have
-4 +/- Square Root[16-4*1*(-1) = =4 +/- Square Root[20]
You can simplifly Square Root[20]=Square Root[4]*Square Root[[5]=Square[2^2]*Square[5]=2*Square Root[5]
The the x-intercepts are (-4+2*Square[5])/2 = -2+Square Root[5] AND
-2-Square Root[5].
In both cases, I was able to cancel 2 in the ny=umerator and denominator.
Learning how to simplify the Sqyare Root of a number was probably one of the objectuves of this problem along with practice using the quadatic formula. Also, don't forget that a negative number times anegative number is a positive number.

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This is a problem involving similar triangles.
The ratios of corresponding sides are equal. Since the ratio of the lengths of the poles is 72/24 = 3 the ratio of the shadows is also 3 = length of longer shadow divided by 8. Therefore the length of the longer shadow is 8*3=24 feet.

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First convert 60% to a decimal number. 60% = .6
The word problem translated into an algebraic equation is
12 = .6*x where is the number you are looking for. THINK ABOUT THIS SO THAT YOU CAN SOLVE FUTURE PROBLEMS.
Next divide both sides of the equation by .6
You now have 12/.6 = x but 12/.6 = 20
so x = 20. Check your answer by multiplyimg 20 by .6

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Rearrange the first equation so that n=13-3m
SUBSTITUTE the rearranged equation into the second equation for n.
The second equation then becomes m-2(13-3m)=23
Use the DISTRIBUTIVE property so that this equation becomes
m-2*13+2*3m = 23 (REMEMBER - negative times a negative is a negative)
We now have m-26+6m = 23
The next step is to combine like terms
We now have 7m - 26 = 23
Add 26 to each side of the equation so that
7m = 49. Divide both sides of the equation by 7 so that m = 7
Substitute this value for m into the first equation to solve for n.
We now have 3*7+n=13
Therefore n = 13-3*7= 13-21 = -8
So m = 7 and n = -8

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The slope of the desired is -(1/2) since the desired line is perpendicular to the given line
So the equation of the desired line in slope intercept form is
y = -(1/2)x+b
Since the point (-2,3) is on the line, this information can be used to solve the equation for "b"
3=-(1/2)(-2)+b (Multiply -(1/2) times (-2) and then solve for b
So b = 1
So the equation is y = -(1/2)x+1

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Since the definition of "average" is ave = sum of all scores divided by the number of scores, we have 80 =(70+86+78+fourth exaximation score)/4
So 320 = 234 + fourth examination score (multiply both sides of the equation by 4 and add the known scores)
So 320-234 = fourth examination score (subtract 234 from each side of the equation
so fourth examination must be greater than or equal to 86 in order to have an average of at 80.

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This question is asking you to substitute 0 for X and 5 for y in both equations. If the right hand value is equal to the left hand value after making the substitution, the answer is "yes". Otherwise, the answer is "no".
Substitute 0 for x and 5 for y in the first equation and then the second equation. Since 0 + 5 = 5 and -5*0 + 2*5 = 10 the answer is "yes".

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Substitute the "value" for y from the first equation into the second equation.
The second equation then becomes 3x+2(x+2)=9
ASp;ve the after using the DISTRIBUTIVE PROPERTY, the second equation becomes 3x+2x+4=9
Solve 3x+2x+4=9 for x
Subtract 4 from each side of the equation
The equation is now 3x+2x=5
Combine like terms so 5x=5
Divide each side of the equation by 5 so x = 1
To solve the system of equations for the other variable "y", replace "x" with 1 in he first equation, so y = 1 + 2 = 3.
So, x = 1 and y = 3.

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Determine 2 points with each point consisting of a x-value and a y-value.
When x = 0, y = -0+6=6
When x = 1, y = -1+6=5
Plot the two points and draw a line running through the two points.

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The formula for the area of a rectangle is A=l*w.
So, the area is 3.5*4.5 = 15.75 meters.
You are asked to express your answer in inches. Presumably, your teacher or your textbook contains a factor that converts meters to inches. THe conversion factor to two decimal places is 1 meter = 39.37 inches. Check to see if you were given a different conversion factor.
15.75 meters * 39.37 inches per meter = 620.078 inches.

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Inequality are solved in a manner just like equalities. You use arithmetic operations (addition, subtraction, multiplication and division) to get the unknown (in this "x")on one side of the inequality and the terms without an unknown on the other side of the inequality.
So,
0.9x+3 < 1.7x-4
Subtract 0.9x from each side of the inequality.
The inequality then reads 3 < 0.8x-4
Add 4 to each side of the inequality.
The inequality then reads 7 < .8x
Divide each side of the inequality by .8
The inequality then reads 7/.8 < x.
Then 70/8 < x

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3000 feet/25 seconds = 120 feet per second

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The growth in population is 50,000-40,000=10,000.
Express the growth in popupation compared to the starting population, i.e. 10,000/40,000. Reduce this fraction to lowest terms. 10,000/40,000=1/4.
Convert 1/4 to a decimal by dividing 1 by 4. 1/4 = .25.
Convert .25 to percent by multipying by 100 and putting a percent sign after the number.
So 1/4 = .25 = 25%

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The calculation is performed by following the "order of precedence of operations". First, do the calculaion inisde of the inner parenthesis, so -2+18=16. so the expression is now 2^3-10*(4-16*3)
Next, multiply 16 times 3 remembering that a negative sign is in fron of the 16.
The expression is now 2^3-10
Now do the calculation inside of the parenthesis, remembering the sign of the answer.
The expression is now 2^3-10*(-44)
Now multiply -10 times -44 (A negative number times a negative number is a positive number. So the expression is now 2^3+440
But 2^3=8
So we now have 8+440 and 8+440 = 448

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The first step is get all the terms of the expression with a "p" on one side of the expression and the other terms on the other side.
So, 8p+3w = w -15-2p can be rewitten as 8p+2p=w-3w-15.
Then combine "like" terms. The equation then becomes, 10p= -2w-15.
Since you are being asked to solve for "p", divide both sides of the equation by 10, Then p = (-2w-15)/10

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Since the triangle is a right triangle, the hypoteneuse of the right triangle will be the longest side of the triangle. You are looking for the shortest side of the triangle. The Pythagorean Theorem says that c^2 = a^2 + b^2, where c is the length of the hypoteneuse. In this case, c=8 and you are looking for b. In this problem, it doesn't matter whether you call the shortest side "a" or "b", all you know for sure is that the shortest side isn't "c".
Therefore, 8^2=6^2+b^2 or 64=36+b^2. So,
b^2=64-36=28. Therefore b = Square Root of 28. rounded to the nearest hundreth is 5.29.

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First, find two pairs of points that satisfy the given equation.
point 1 -- When x = 0, y = -3*0=0. So (0,0) is one point on the graph of the equation.
point 2 -- When x = 1, y = -3*1=-1, So (1,-3) is another point on the graph of the equation.
You can plot more points but two points determine the graph of a line.

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Let l = the length of the rectangle.
Let w = the width of the rectangle.
l = 2 + w.
The perimeter of rectangle is 2*length + 2*width and in this problem the perimeter is 20.
Putting this information into an algebraic equation, you have 2*l+2*w = 20. Now you know that l = 2 + w so you can substitue this expression for "l" into the equation for the perimeter. This gives you, 2*(2+w)+2*w=20.
Use the Distributive Property to simplify the equation so you now have
4+2*w+2*w=20. Combining like terms, you have 4+4*w=20. Since 4 is a factor of each term in the equation, you can divide each term of the equation by 4 and you now have 1 + w = 5. Subtract 1 from each side of the equation and you have w=4. Since l = 2+w, l=6.
Therefore the width of the rectangle is 4 feet and the length is 6 feet.
Have fun studying math.
Larry

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Using the distance formula, fund the length of each of the 3 ides of the triangle,
Length of side 1 --- Square Root of [(3-(-3))^2+(10-8)^2]=Square Root of [6^2+2^2]=Square Root of 40.
Length of Side 2 --- Square Root of [7-3)^2+(-2-10)^2]=Square Root of [4^2+(-12)^2]=Square Root of [16+144] =Square Root of 160.
Length of Side 3 --- Square Root of [-3-7)^2+(8-(-2))^2]=Square Root of [(-10)^2+(8+2)^2]=Square Root of [100+100] =Square Root of 200.
Since the length of side 3 is the largest it must be the hypoteneuse of the right triangle if the triangle is to be a right triangle. For the triangle to be a right triangle, the Pythagorean Theorem has to be met so that
200^2 must be equal 160^2 + 40^2. In other words, 200^2=40^2+160^2. But this isn't true, so the triangle is NOT a right triangle.
The problem is an attempt to get you to show that something isn't true by using facts that must be true if the original statement was true.
Have fun studying math.

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Let n = the number you are interested in finding. The description of the problem says "5 less than the number is 15". The key is to find the important words such as "is less than" and "is". "Is less than" means "subtraction" and "is" mean "= (equals)" Rewriting the words as an algebraic equation you have n-5=15. Using the "Rules of algebra", add 5 to each side of the equation, s0 n-5+5=15+5. So n=20.
Good luck studying math.
Larry

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PROBLEM 1
Let n = the first integer, so n+1 is the next integer. Now translate the word problem into an algebraic expresion. So, (n+(n+1))^2=361. Take the square root of both sides of the equation, so, n+(n+1)=19 and n+(n+1)=-19.
Case 1 yields the following linear equation 2n+1=19. Solving this linear equation results in the value n = (19-1)/2=9. The pair is 9 and 10
Case 2 yields the folowing linear equation n+(n+1)=-19. Solving this linear equation results in the value for n = (-19-1)/2=-10. The pair is then -10
and -9.
Problem 2
Let n = the first even integer. So n = 2k for some k. The next even integer is then n+2 = 2k+2. Therefore, the words of the problem can be translated into the following algebraic equation 2k*(2k+2)=288. Expanding this equation we have 4k^2+4k-288=0. Dividing both sides by 4 produces k^2+k-72=0. This equation can be solved for k by factoring. k^2+k-72=(k+9)(k-8)=0. Therefore,
k=-9 and k=8. So, the first pair of integers is n=2*k = -18 and the second integer is -16.
The second pair of integers is 2*8=16 and 18.
It is important to recognize the pairs of solutions for each question.
Good luck with your study of math.
Larry

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I'm assuming that the question is:
Factor (4x^3-25x)/(x^2-14x+48)
4x^3-25x= x*(4x^2-25)=x*(2x+5)*(2x-5)
and
x^2-14x+48=(x-8)(x-6)
Therefore,
(4x^3-25x)/(x^2-14x+48)=x*((2x+5)*(2x-5))/((x-8)(x-6))
Good luck with your math studies.

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Let x = length of the original square
So (x+8)^2=289
Take the square root of each side of the equation.
So x+8 = 17 and -17
Solve each linear equation
x+8=17, so x=9
and x+8=-17, so x = -25.
Since I believe that the question is dealing with a real, physical square, the second answer (x=-25) is dropped and the answer is x=9 cm.
Good luck with your math studies.

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This problem is basically the inverse of being asked to factor a quadratic equation.
The solution (roots) of the equation are 8 and 8. Ues, 8 is a repeating root since the goal is to produce an equation in standard form of degree 2. An equation of degree 2 has 2 solutions therefore 8 is a repeating solution. If "a" is a solution (root) of an equation then "(x-a)" is a factor. So, in this case, the equation that you are being asked to produce has factors "(x-8)" and "(x-8)". Multiply the the factors using the "FOIL" method.
Therefore, the equation that you are being asked to produce is
(x-8)(x-8)= x^2-16x+64=0.

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A quadratic equation may have 0,1,or 2 REAL solutions. REAL solutions refers to real numbers as compared to COMPLEX numbers.
To determine the number of REAL solutions, you need to calculate the amount under the radical arising from the application of the quadratic formula, i,e b^2-4ac.
b^2-4ac is called the determinant. If the determinant = 0, the quadratic equation has one REAL solution. If the determinant > 0, the quadratic equation has 2 REAL solutions. If the determinant < 0, there are 0 REAL solutions.
Larry

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The absolute value of -6 is 6.
The absolute value of -2 is 2.
So the absolute value of -6 minus the absolute value of -2 is 6 - 2 = 4.
Larry

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The function is in slope intercept form. So the slope is the coefficient of the x term. So the slope is -8. To find the y-intercept, substitute 0 for x in the function. So when x = 0, y = -8*0-10 = -10. The y-intercept is (0,-10).
Larry