Given the system of linear equations.
{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.
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:
