The graph of a linear equation passes through the points (-4, 1) and (4, 6). Select all the points that are also solutions to this equation.

Steps to Solve

1. Calculate the Slope of the Line To find the slope $m$ of the line passing through the points $(-4, 1)$ and $(4, 6)$, use the formula: $$ m = \frac{y_2 - y_1}{x_2 - x_1} $$ Here, $(x_1, y_1) = (-4, 1)$ and $(x_2, y_2) = (4, 6)$. $$ m = \frac{6 - 1}{4 - (-4)} = \frac{5}{8} $$

2. Find the Equation of the Line Using the slope-intercept form $y = mx + b$, we can find the y-intercept $b$ using one of the points. Let's use $(-4, 1)$: $$ 1 = \left(\frac{5}{8}\right)(-4) + b $$ Solving for $b$: $$ 1 = -\frac{20}{8} + b $$ $$ b = 1 + \frac{20}{8} = 1 + 2.5 = 3.5 $$ Thus, the equation of the line is: $$ y = \frac{5}{8}x + 3.5 $$

3. Check Each Given Point We will now substitute each given point into the equation to see if it satisfies it.

• For $(0, 3.5)$: $$ 3.5 = \frac{5}{8}(0) + 3.5 \quad \text{(True)} $$

• For $(8, 5)$: $$ 5 = \frac{5}{8}(8) + 3.5 $$ $$ 5 = 5 + 3.5 \quad \text{(False)} $$

• For $(12, 11)$: $$ 11 = \frac{5}{8}(12) + 3.5 $$ $$ 11 = 7.5 + 3.5 \quad \text{(True)} $$

• For $(-6, 0)$: $$ 0 = \frac{5}{8}(-6) + 3.5 $$ $$ 0 = -3.75 + 3.5 \quad \text{(False)} $$

• For $(5, 6)$: $$ 6 = \frac{5}{8}(5) + 3.5 $$ $$ 6 = 3.125 + 3.5 \quad \text{(False)} $$

4. List of Valid Points The points that satisfy the equation are: $(0, 3.5)$ and $(12, 11)$.