Solving Equations to Find the Values of x and y

When faced with the equations y x - 9 and 2x - 2y 30, how can we determine the values of x and y? In this article, we’ll walk through the process of solving these equations step by step.

Introduction to the Equations

We are given two linear equations:
y x - 9
2x - 2y 30 To solve for x and y, we can use the method of substitution. First, let's express y in terms of x from the first equation.

Step 1: Express y in Terms of x

Given y x - 9, we can substitute this expression directly into our second equation.

Step 2: Substitute and Simplify

Substitute y x - 9 into the second equation 2x - 2y 30:

Replace y with x - 9:
2x - 2(x - 9) 30 Simplify the equation by distributing the 2:
2x - 2x 18 30 Combine like terms:
18 30 - 2x Isolate the x term:
4x 12 Solve for x:
x 12 / 4 3

Step 3: Find the Value of y

Now that we have the value of x, we can substitute it back into the first equation to solve for y.

Substitute x 3 into y x - 9:
y 3 - 9 -6

Thus, the values are:
x 3 and y -6.

Verification of the Solution

To verify our solution, we can substitute the values of x and y back into the original equations:

Equation 1: y x - 9
nMotion for x 3 and y -6:

-6 3 - 9

-6 -6

Equation 2: 2x - 2y 30
nMotion for x 3 and y -6:

2 * 3 - 2 * (-6) 30

6 12 30

18 18

These checks confirm that the solution is correct.

Graphical Interpretation

The two equations can also be graphed to visualize their intersection point. The line y x - 9 forms a downward slant, while the line 2x - 2y 30 simplifies to y x - 15 when converted into slope-intercept form. The point (3, -6) is where these lines intersect, confirming our algebraic solution.

Conclusion

Through the method of substitution, we found that the solution to the system of equations y x - 9 and 2x - 2y 30 is x 3 and y -6. This solution satisfies both given equations, as verified by substitution and graphical interpretation.