SOLUTION: Solve the system of linear equations using the substitution method: -4x+y=7 y=-3x+6

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Question 161997: Solve the system of linear equations using the substitution method:
-4x+y=7
y=-3x+6
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

-4x%2By=7
y=-3x%2B6 or 3x%2By=6

Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

-4%2Ax%2B1%2Ay=7
3%2Ax%2B1%2Ay=6

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

1%2Ay=7%2B4%2AxAdd 4%2Ax to both sides

y=%287%2B4%2Ax%29 Divide both sides by 1.


Which breaks down and reduces to



y=7%2B4%2Ax Now we've fully isolated y

Since y equals 7%2B4%2Ax we can substitute the expression 7%2B4%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


3%2Ax%2B1%2Ahighlight%28%287%2B4%2Ax%29%29=6 Replace y with 7%2B4%2Ax. Since this eliminates y, we can now solve for x.

3%2Ax%2B1%2A%287%29%2B1%284%29x=6 Distribute 1 to 7%2B4%2Ax

3%2Ax%2B7%2B4%2Ax=6 Multiply



3%2Ax%2B7%2B4%2Ax=6 Reduce any fractions

3%2Ax%2B4%2Ax=6-7 Subtract 7 from both sides


3%2Ax%2B4%2Ax=-1 Combine the terms on the right side



7%2Ax=-1 Now combine the terms on the left side.


cross%28%281%2F7%29%287%2F1%29%29x=%28-1%2F1%29%281%2F7%29 Multiply both sides by 1%2F7. This will cancel out 7%2F1 and isolate x

So when we multiply -1%2F1 and 1%2F7 (and simplify) we get



x=-1%2F7 <---------------------------------One answer

Now that we know that x=-1%2F7, lets substitute that in for x to solve for y

3%28-1%2F7%29%2B1%2Ay=6 Plug in x=-1%2F7 into the 2nd equation

-3%2F7%2B1%2Ay=6 Multiply

1%2Ay=6%2B3%2F7Add 3%2F7 to both sides

1%2Ay=42%2F7%2B3%2F7 Make 6 into a fraction with a denominator of 7



1%2Ay=45%2F7 Combine the terms on the right side

cross%28%281%2F1%29%281%29%29%2Ay=%2845%2F7%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1 on the left side.

y=45%2F7 Multiply the terms on the right side


y=45%2F7 Reduce


So this is the other answer


y=45%2F7<---------------------------------Other answer


So our solution is

x=-1%2F7 and y=45%2F7

which can also look like

(-1%2F7,45%2F7)

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

-4%2Ax%2B1%2Ay=7
3%2Ax%2B1%2Ay=6

we get


graph of -4%2Ax%2B1%2Ay=7 (red) and 3%2Ax%2B1%2Ay=6 (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%2F7,45%2F7). This verifies our answer.


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Check:

Plug in (-1%2F7,45%2F7) into the system of equations


Let x=-1%2F7 and y=45%2F7. Now plug those values into the equation -4%2Ax%2B1%2Ay=7

-4%2A%28-1%2F7%29%2B1%2A%2845%2F7%29=7 Plug in x=-1%2F7 and y=45%2F7


4%2F7%2B45%2F7=7 Multiply


49%2F7=7 Add


7=7 Reduce. Since this equation is true the solution works.


So the solution (-1%2F7,45%2F7) satisfies -4%2Ax%2B1%2Ay=7



Let x=-1%2F7 and y=45%2F7. Now plug those values into the equation 3%2Ax%2B1%2Ay=6

3%2A%28-1%2F7%29%2B1%2A%2845%2F7%29=6 Plug in x=-1%2F7 and y=45%2F7


-3%2F7%2B45%2F7=6 Multiply


42%2F7=6 Add


6=6 Reduce. Since this equation is true the solution works.


So the solution (-1%2F7,45%2F7) satisfies 3%2Ax%2B1%2Ay=6


Since the solution (-1%2F7,45%2F7) satisfies the system of equations


-4%2Ax%2B1%2Ay=7
3%2Ax%2B1%2Ay=6


this verifies our answer.