Question

Solve the system of linear equations by graphing.  

x minus 2 yx−2y
​=
44
3 x plus 2 y3x+2y
​=
negative 4−4

Answer 1

To solve the system of equations by graphing, we start by rewriting both equations in slope-intercept form (\(y = mx + b\)).

The given equations are:

1. \(x - 2y = 44\)
2. \(3x + 2y = -4\)

### Step 1: Convert the equations to slope-intercept form

#### For the first equation: \(x - 2y = 44\)

1. Isolate \(y\):
\[
-2y = -x + 44
\]
\[
y = \frac{1}{2}x - 22
\]

#### For the second equation: \(3x + 2y = -4\)

1. Isolate \(y\):
\[
2y = -3x - 4
\]
\[
y = -\frac{3}{2}x - 2
\]

### Step 2: Graph both equations

1. **Graph \(y = \frac{1}{2}x - 22\)**:
- The y-intercept is \(-22\).
- The slope is \(\frac{1}{2}\), which means for every increase of 1 in \(x\), \(y\) increases by \(\frac{1}{2}\).

2. **Graph \(y = -\frac{3}{2}x - 2\)**:
- The y-intercept is \(-2\).
- The slope is \(-\frac{3}{2}\), which means for every increase of 1 in \(x\), \(y\) decreases by \(\frac{3}{2}\).

### Step 3: Find the intersection of the two lines

To determine where these lines intersect, you can look for a point on the graph where both equations equal the same \(y\) value for the same \(x\) value. Alternatively, you can set the two equations equal to find the \(x\)-coordinate of the intersection point.

Set the two equations equal:
\[
\frac{1}{2}x - 22 = -\frac{3}{2}x - 2
\]

### Step 4: Solve for \(x\)

1. Combine like terms:
\[
\frac{1}{2}x + \frac{3}{2}x = 22 - 2
\]
\[
2x = 20 \quad (Multiply both sides by 2)
\]
\[
x = 10
\]

### Step 5: Solve for \(y\)

Substitute \(x = 10\) back into one of the original equations. We can use the first equation:
\[
y = \frac{1}{2}(10) - 22 = 5 - 22 = -17
\]

### Step 6: Conclusion

The solution to the system of equations is:
\[
(x, y) = (10, -17)
\]

You would graph both equations and confirm visually that they intersect at the point \((10, -17)\).