SOLUTION: Solve using either elimination or substitution. Show your work. If the system has either no solution or an infinite number of solutions, state this. 16x-15y=-10 11x-7y=13

Algebra ->  Coordinate Systems and Linear Equations -> SOLUTION: Solve using either elimination or substitution. Show your work. If the system has either no solution or an infinite number of solutions, state this. 16x-15y=-10 11x-7y=13      Log On


   

Question 295557: Solve using either elimination or substitution. Show your work. If the system has either no solution or an infinite number of solutions, state this.
16x-15y=-10
11x-7y=13
Answer by edjones(8007) About Me  (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

16%2Ax-15%2Ay=-10
11%2Ax-7%2Ay=13

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

-15%2Ay=-10-16%2AxSubtract 16%2Ax from both sides

y=%28-10-16%2Ax%29%2F-15 Divide both sides by -15.


Which breaks down and reduces to



y=2%2F3%2B%2816%2F15%29%2Ax Now we've fully isolated y

Since y equals 2%2F3%2B%2816%2F15%29%2Ax we can substitute the expression 2%2F3%2B%2816%2F15%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


11%2Ax%2B-7%2Ahighlight%28%282%2F3%2B%2816%2F15%29%2Ax%29%29=13 Replace y with 2%2F3%2B%2816%2F15%29%2Ax. Since this eliminates y, we can now solve for x.

11%2Ax-7%2A%282%2F3%29-7%2816%2F15%29x=13 Distribute -7 to 2%2F3%2B%2816%2F15%29%2Ax

11%2Ax-14%2F3-%28112%2F15%29%2Ax=13 Multiply



11%2Ax-14%2F3-%28112%2F15%29%2Ax=13 Reduce any fractions

11%2Ax-%28112%2F15%29%2Ax=13%2B14%2F3Add 14%2F3 to both sides


11%2Ax-%28112%2F15%29%2Ax=39%2F3%2B14%2F3 Make 13 into a fraction with a denominator of 3


11%2Ax-%28112%2F15%29%2Ax=53%2F3 Combine the terms on the right side



%28165%2F15%29%2Ax-%28112%2F15%29x=53%2F3 Make 11 into a fraction with a denominator of 15

%2853%2F15%29%2Ax=53%2F3 Now combine the terms on the left side.


cross%28%2815%2F53%29%2853%2F15%29%29x=%2853%2F3%29%2815%2F53%29 Multiply both sides by 15%2F53. This will cancel out 53%2F15 and isolate x

So when we multiply 53%2F3 and 15%2F53 (and simplify) we get



x=5 <---------------------------------One answer

Now that we know that x=5, lets substitute that in for x to solve for y

11%285%29-7%2Ay=13 Plug in x=5 into the 2nd equation

55-7%2Ay=13 Multiply

-7%2Ay=13-55Subtract 55 from both sides

-7%2Ay=-42 Combine the terms on the right side

cross%28%281%2F-7%29%28-7%29%29%2Ay=%28-42%2F1%29%281%2F-7%29 Multiply both sides by 1%2F-7. This will cancel out -7 on the left side.

y=-42%2F-7 Multiply the terms on the right side


y=6 Reduce


So this is the other answer


y=6<---------------------------------Other answer


So our solution is

x=5 and y=6

which can also look like

(5,6)

Notice if we graph the equations (if you need help with graphing, check out this solver)

16%2Ax-15%2Ay=-10
11%2Ax-7%2Ay=13

we get


graph of 16%2Ax-15%2Ay=-10 (red) and 11%2Ax-7%2Ay=13 (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 (5,6). This verifies our answer.


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Check:

Plug in (5,6) into the system of equations


Let x=5 and y=6. Now plug those values into the equation 16%2Ax-15%2Ay=-10

16%2A%285%29-15%2A%286%29=-10 Plug in x=5 and y=6


80-90=-10 Multiply


-10=-10 Add


-10=-10 Reduce. Since this equation is true the solution works.


So the solution (5,6) satisfies 16%2Ax-15%2Ay=-10



Let x=5 and y=6. Now plug those values into the equation 11%2Ax-7%2Ay=13

11%2A%285%29-7%2A%286%29=13 Plug in x=5 and y=6


55-42=13 Multiply


13=13 Add


13=13 Reduce. Since this equation is true the solution works.


So the solution (5,6) satisfies 11%2Ax-7%2Ay=13


Since the solution (5,6) satisfies the system of equations


16%2Ax-15%2Ay=-10
11%2Ax-7%2Ay=13


this verifies our answer.