SOLUTION: solve using substitution 2x + 7y = 12 and 5x - 4y = -13

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Question 192018: solve using substitution 2x + 7y = 12 and 5x - 4y = -13
Answer by vleith(2983) About Me  (Show Source):
You can put this solution on YOUR website!
Use this tool --> http://www.algebra.com/algebra/college/linear/solving-linear-system-by-substitution.solver as follows
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

2%2Ax%2B7%2Ay=12
5%2Ax-4%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

7%2Ay=12-2%2AxSubtract 2%2Ax from both sides

y=%2812-2%2Ax%29%2F7 Divide both sides by 7.


Which breaks down and reduces to



y=12%2F7-%282%2F7%29%2Ax Now we've fully isolated y

Since y equals 12%2F7-%282%2F7%29%2Ax we can substitute the expression 12%2F7-%282%2F7%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


5%2Ax%2B-4%2Ahighlight%28%2812%2F7-%282%2F7%29%2Ax%29%29=-13 Replace y with 12%2F7-%282%2F7%29%2Ax. Since this eliminates y, we can now solve for x.

5%2Ax-4%2A%2812%2F7%29-4%28-2%2F7%29x=-13 Distribute -4 to 12%2F7-%282%2F7%29%2Ax

5%2Ax-48%2F7%2B%288%2F7%29%2Ax=-13 Multiply



5%2Ax-48%2F7%2B%288%2F7%29%2Ax=-13 Reduce any fractions

5%2Ax%2B%288%2F7%29%2Ax=-13%2B48%2F7Add 48%2F7 to both sides


5%2Ax%2B%288%2F7%29%2Ax=-91%2F7%2B48%2F7 Make -13 into a fraction with a denominator of 7


5%2Ax%2B%288%2F7%29%2Ax=-43%2F7 Combine the terms on the right side



%2835%2F7%29%2Ax%2B%288%2F7%29x=-43%2F7 Make 5 into a fraction with a denominator of 7

%2843%2F7%29%2Ax=-43%2F7 Now combine the terms on the left side.


cross%28%287%2F43%29%2843%2F7%29%29x=%28-43%2F7%29%287%2F43%29 Multiply both sides by 7%2F43. This will cancel out 43%2F7 and isolate x

So when we multiply -43%2F7 and 7%2F43 (and simplify) we get



x=-1 <---------------------------------One answer

Now that we know that x=-1, lets substitute that in for x to solve for y

5%28-1%29-4%2Ay=-13 Plug in x=-1 into the 2nd equation

-5-4%2Ay=-13 Multiply

-4%2Ay=-13%2B5Add 5 to both sides

-4%2Ay=-8 Combine the terms on the right side

cross%28%281%2F-4%29%28-4%29%29%2Ay=%28-8%2F1%29%281%2F-4%29 Multiply both sides by 1%2F-4. This will cancel out -4 on the left side.

y=-8%2F-4 Multiply the terms on the right side


y=2 Reduce


So this is the other answer


y=2<---------------------------------Other answer


So our solution is

x=-1 and y=2

which can also look like

(-1,2)

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

2%2Ax%2B7%2Ay=12
5%2Ax-4%2Ay=-13

we get


graph of 2%2Ax%2B7%2Ay=12 (red) and 5%2Ax-4%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 (-1,2). This verifies our answer.


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Check:

Plug in (-1,2) into the system of equations


Let x=-1 and y=2. Now plug those values into the equation 2%2Ax%2B7%2Ay=12

2%2A%28-1%29%2B7%2A%282%29=12 Plug in x=-1 and y=2


-2%2B14=12 Multiply


12=12 Add


12=12 Reduce. Since this equation is true the solution works.


So the solution (-1,2) satisfies 2%2Ax%2B7%2Ay=12



Let x=-1 and y=2. Now plug those values into the equation 5%2Ax-4%2Ay=-13

5%2A%28-1%29-4%2A%282%29=-13 Plug in x=-1 and y=2


-5-8=-13 Multiply


-13=-13 Add


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


So the solution (-1,2) satisfies 5%2Ax-4%2Ay=-13


Since the solution (-1,2) satisfies the system of equations


2%2Ax%2B7%2Ay=12
5%2Ax-4%2Ay=-13


this verifies our answer.