Question

Choose the graph that solves this system of equations.

y = 2x – 6
y = –4x + 3

Answer 1

To solve the system of equations given by:

1. \( y = 2x - 6 \)
2. \( y = -4x + 3 \)

we can graph both equations and find their intersection.

### Step 1: Graph the First Equation
For \( y = 2x - 6 \):
- The y-intercept is at \( (0, -6) \)
- The slope is 2, which means for every 1 unit increase in \( x \), \( y \) increases by 2 units.

**Points on the line:**
- When \( x = 0 \), \( y = -6 \) → Point (0, -6)
- When \( x = 3 \), \( y = 2(3) - 6 = 0 \) → Point (3, 0)
- When \( x = 1 \), \( y = 2(1) - 6 = -4 \) → Point (1, -4)

### Step 2: Graph the Second Equation
For \( y = -4x + 3 \):
- The y-intercept is at \( (0, 3) \)
- The slope is -4, which means for every 1 unit increase in \( x \), \( y \) decreases by 4 units.

**Points on the line:**
- When \( x = 0 \), \( y = 3 \) → Point (0, 3)
- When \( x = 1 \), \( y = -4(1) + 3 = -1 \) → Point (1, -1)
- When \( x = -1 \), \( y = -4(-1) + 3 = 7 \) → Point (-1, 7)

### Step 3: Find the Intersection Point
To find the intersection of the two lines, we can set the equations equal to each other:

\[ 2x - 6 = -4x + 3 \]

Now solve for \( x \):
\[ 2x + 4x = 3 + 6 \]
\[ 6x = 9 \]
\[ x = \frac{9}{6} = \frac{3}{2} \]

Now substitute \( x = \frac{3}{2} \) back into one of the original equations to solve for \( y \):
Using \( y = 2x - 6 \):
\[ y = 2\left(\frac{3}{2}\right) - 6 \]
\[ y = 3 - 6 \]
\[ y = -3 \]

Thus, the intersection point is \( \left(\frac{3}{2}, -3\right) \).

### Conclusion
The correct graph will show the lines intersecting at the point \( \left(\frac{3}{2}, -3\right) \). You can check multiple graphs, and look for the one with that intersection point to identify the solution for the system of equations.