Solve this system of equations by graphing. First graph the equations, and then type the solution.

x–y=2
x–4y=

4
Click to select points on the graph.

1 answer

To solve the system of equations by graphing, let's write down both equations clearly first. The given equations are:

  1. \( x - y = 2 \)
  2. \( x - 4y = -4 \)

Step 1: Rewrite the Equations in Slope-Intercept Form

To graph these equations, we'll rewrite them in the form \( y = mx + b \).

  1. First Equation: \[ x - y = 2 \] Rearranging gives: \[ y = x - 2 \]

  2. Second Equation: \[ x - 4y = -4 \] Rearranging gives: \[ 4y = x + 4 \quad \Rightarrow \quad y = \frac{1}{4}x + 1 \]

Step 2: Graph the Equations

First Equation: \( y = x - 2 \)

  • This line has a y-intercept of -2 and a slope of 1.
  • Two points you can graph for this line:
    • When \( x = 0 \): \( y = 0 - 2 = -2 \) → (0, -2)
    • When \( x = 2 \): \( y = 2 - 2 = 0 \) → (2, 0)

Second Equation: \( y = \frac{1}{4}x + 1 \)

  • This line has a y-intercept of 1 and a slope of \(\frac{1}{4}\).
  • Two points you can graph for this line:
    • When \( x = 0 \): \( y = \frac{1}{4}(0) + 1 = 1 \) → (0, 1)
    • When \( x = 4 \): \( y = \frac{1}{4}(4) + 1 = 2 \) → (4, 2)

Step 3: Graphing

You would take these points and draw each line on a graph.

Intersection Point: The solution to the system of equations is where the two lines intersect.

Step 4: Finding the Intersection Point

To find the coordinates of the intersection point algebraically: Set the two equations equal to each other: \[ x - 2 = \frac{1}{4}x + 1 \]

Now, solve for \(x\):

  1. Subtract \( \frac{1}{4}x \) from both sides: \[ x - \frac{1}{4}x = 2 + 1 \] \[ \frac{3}{4}x = 3 \]

  2. Multiply both sides by \( \frac{4}{3} \): \[ x = 4 \]

Now, substitute \(x = 4\) back into either equation to find \(y\): Using \(y = x - 2\): \[ y = 4 - 2 = 2 \]

Final Solution

The solution to the system of equations is the point where the two lines intersect: \[ \boxed{(4, 2)} \]

You can verify by plugging these values back into both original equations.