Finish solving the system of equations X – 5y = 6 and -x + 2y = -3. What is the value of y? Substitute the value of y back into one of the original equations to find the value of x. What is the value of x?
The Correct Answer and Explanation is:
We are given a system of equations:
1) x – 5y = 6
2) -x + 2y = -3
Step 1: Add the two equations
We observe that if we add both equations together, the x terms will cancel out:(x−5y)+(−x+2y)=6+(−3)(x – 5y) + (-x + 2y) = 6 + (-3)(x−5y)+(−x+2y)=6+(−3)
Simplify:x−5y−x+2y=3x – 5y – x + 2y = 3x−5y−x+2y=3−3y=3-3y = 3−3y=3
Step 2: Solve for y
Divide both sides by -3:y=−1y = -1y=−1
Step 3: Substitute y = -1 into one of the original equations
We will use the first equation:x−5y=6x – 5y = 6x−5y=6
Substitute y = -1:x−5(−1)=6x – 5(-1) = 6x−5(−1)=6x+5=6x + 5 = 6x+5=6x=1x = 1x=1
Final Answer:
- y = -1
- x = 1
Explanation
To solve a system of linear equations, one effective method is the elimination method, which involves combining the equations to eliminate one variable. In the given system:
- x – 5y = 6
- -x + 2y = -3
We notice that the coefficients of x are opposites (x and -x). This makes it easy to eliminate x by adding the two equations. Doing this step-by-step:
(x – 5y) + (-x + 2y) results in -3y on the left side and 3 on the right side, so we get the equation:
-3y = 3.
Dividing both sides by -3 gives y = -1.
Once we have the value of y, we can substitute it back into either of the original equations to find the corresponding value of x. Using the first equation:
x – 5y = 6 becomes x – 5(-1) = 6. This simplifies to x + 5 = 6, and solving for x gives x = 1.
This process confirms the solution is the ordered pair (1, -1). If you plug these values into the second original equation:
-x + 2y = -3 becomes -1 + 2(-1) = -3, which is true.
Thus, both x and y satisfy the system. The solution is correct.
