To solve the system of equations by graphing, let's write down both equations clearly first. The given equations are:
- \( x - y = 2 \)
- \( x - 4y = -4 \)
Step 1: Rewrite the Equations in Slope-Intercept Form
To graph these equations, we'll rewrite them in the form \( y = mx + b \).
-
First Equation: \[ x - y = 2 \] Rearranging gives: \[ y = x - 2 \]
-
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 \(x\):
-
Subtract \( \frac{1}{4}x \) from both sides: \[ x - \frac{1}{4}x = 2 + 1 \] \[ \frac{3}{4}x = 3 \]
-
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.