Question

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.

Answer 1

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.