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Question 692992: What is the solution of the system? Use substitution.
2x - y = -7
4x - y = -4
Answer by RedemptiveMath(80)   (Show Source):
You can put this solution on YOUR website! We can do this problem four ways. Each way will give the same answer as the other three. When we want to solve a system of linear equations using substitution, we need to rearrange one of the equations we are given so that a variable is by itself. In other words, we need to solve for one of the variables by algebraically manipulating one of the equations. I will attempt to do one way below.
4x - y = -4
-y = -4 - 4x (subtract 4x from each side)
y = 4 + 4x (divide by -1)
y = 4x + 4 (commutative property of addition to get the x-value in front).
Now we must plug what we have gotten y to equal into the other equation. If we try to plug what we have rearranged back into the original equation, we don't get anywhere.
2x - y = -7
2x - 1(4x + 4) = -7 (substitution)
2x - 4x - 4 = -7 (distributive property with -1)
-2x - 4 = -7 (combine like terms)
-2x = -3 (add 4 to each side)
x = 3/2 (divide by -2).
We have just solved for x! We can now plug what we have found for x into either equation to get what y equals. (If this is truly a system of linear equations, then it will not matter which one we use.)
2(3/2) - y = -7 (substitution)
6/2 - y = -7 (multiplication)
3 - y = -7 (division)
- y = -10 (subtraction)
y = 10 (division)
We have just found y! As a point, the solution to this system is (3/2, 10) with the x-coordinate first and the y-coordinate second. If we plug x and y into both equations, we will find that they will make the equations true. This is how we check ourselves. Using both equations:
4x - y = -4
4(3/2) - 10 = -4 (does the left half equal -4?)
12/2 - 10 = -4
6 - 10 = -4
-4 = -4 (it does);
2x - y = -7
2(3/2) - 10 = -7
3 - 10 = -7
-7 = -7.
Again, we can manipulate either equation for either variable to start off. Then we just plug what we have found for the first variable into the other equation. (Remember that at this point the variable will not equal a constant value because we still have the other variable to deal with.) We simplify and get a constant value for the other variable. Next we plug that value into its variable for either equation and we receive the constant value for the other variable. Finally, to check we just plug both constant values into their variables for both equations and see if they come out to be true.
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