Given the system of linear equations.

Question

{x-y=5
{4x+y=10
Part A: Use substitution to find the solution to the systems of equations. Include all of your work in your final answer.

Part B: Algebraically verify your answer to Part A. Include all of your work in your final answer.

Part C: Graphically verify your answer to Part A. In your final answer, include the graph.

Answer 1

Part A:

From the first equation, we can solve for x:

x = y + 5

Substitute this expression for x into the second equation:

4(y + 5) + y = 10

Distribute the 4:

4y + 20 + y = 10

Combine like terms:

5y + 20 = 10

Subtract 20 from both sides:

5y = -10

Divide both sides by 5:

y = -2

Now that we have the value of y, we can use either of the original equations to solve for x:

x - (-2) = 5

x + 2 = 5

x = 3

Therefore, the solution to the system of equations is (x,y) = (3,-2).

Part B:

To verify our answer, we can substitute (x,y) = (3,-2) into both equations and check that they hold true:

x - y = 3 - (-2) = 5

4x + y = 4(3) + (-2) = 10

Therefore, our answer is algebraically verified.

Part C:

To graphically verify our answer, we can plot the two lines corresponding to the two equations and see where they intersect.

First, rearrange the first equation to solve for y:

y = x - 5

Plot the y-intercept (0,-5) and another point by picking any value for x and finding the corresponding value for y using the equation. For example, when x = 5:

y = 5 - 5 = 0

So we have another point (5,0).

Next, rearrange the second equation to solve for y:

y = 10 - 4x

Plot the y-intercept (0,10) and another point by picking any value for x and finding the corresponding value for y using the equation. For example, when x = 1:

y = 10 - 4(1) = 6

So we have another point (1,6).

The two lines intersect at the point (3,-2), which confirms our answer.

Here is a graph of the two lines: