Question

write the equation of the line in slope - intercept form that passes through (1, -2) and (-4, -5).
show hint
write the equation of the line in slope - intercept form that passes through (-2, 4) and (3, 2).
show hint
which of the following explains how to find the slope of the line that passes through two points? select all that apply.
□ substitute values into the slope - intercept form and solve for b.
□ $m = \frac{y_2 - y_1}{x_2 - x_1}$
□ use the slope - intercept form, $y = mx + b$.
□ graph them and count rise over run.
□ find the y - intercept.

Explanation:

First Sub - Question: Write the equation of the line in slope - intercept form that passes through (1, -2) and (-4, -5).

Step 1: Calculate the slope (m)

The formula for slope between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let \((x_1,y_1)=(1,-2)\) and \((x_2,y_2)=(-4,-5)\). Then \(m=\frac{-5-(-2)}{-4 - 1}=\frac{-5 + 2}{-5}=\frac{-3}{-5}=\frac{3}{5}\).

Step 2: Find the y - intercept (b)

Use the slope - intercept form \(y=mx + b\). Substitute \(m = \frac{3}{5}\), \(x = 1\) and \(y=-2\) into the equation: \(-2=\frac{3}{5}(1)+b\). Solve for \(b\): \(-2-\frac{3}{5}=b\), \(-\frac{10}{5}-\frac{3}{5}=b\), \(b=-\frac{13}{5}\).

Step 3: Write the equation

The slope - intercept form is \(y=mx + b\). Substitute \(m=\frac{3}{5}\) and \(b = -\frac{13}{5}\) into the equation: \(y=\frac{3}{5}x-\frac{13}{5}\).

Second Sub - Question: Write the equation of the line in slope - intercept form that passes through (-2, 4) and (3, 2).

Step 1: Calculate the slope (m)

Using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\), let \((x_1,y_1)=(-2,4)\) and \((x_2,y_2)=(3,2)\). Then \(m=\frac{2 - 4}{3-(-2)}=\frac{-2}{5}=-\frac{2}{5}\).

Step 2: Find the y - intercept (b)

Substitute \(m = -\frac{2}{5}\), \(x=-2\) and \(y = 4\) into \(y=mx + b\): \(4=-\frac{2}{5}(-2)+b\), \(4=\frac{4}{5}+b\). Solve for \(b\): \(b=4-\frac{4}{5}=\frac{20}{5}-\frac{4}{5}=\frac{16}{5}\).

Step 3: Write the equation

Substitute \(m=-\frac{2}{5}\) and \(b=\frac{16}{5}\) into \(y = mx + b\): \(y=-\frac{2}{5}x+\frac{16}{5}\).

Third Sub - Question: Which of the following explains how to find the slope of the line that passes through two points? Select all that apply.

• "Substitute values into the slope - intercept form and solve for b" is used to find the y - intercept, not the slope.
• The formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\) is the direct formula for calculating the slope between two points.
• "Use the slope - intercept form, \(y = mx + b\)" is for writing the equation of a line, not directly for finding the slope.
• "Graph them and count rise over run" is a graphical method to find the slope (rise over run is the definition of slope).
• "Find the y - intercept" is not a method to find the slope.

Answer:

• First sub - question: \(y=\frac{3}{5}x-\frac{13}{5}\)
• Second sub - question: \(y = -\frac{2}{5}x+\frac{16}{5}\)
• Third sub - question: B. \(m=\frac{y_2 - y_1}{x_2 - x_1}\), D. Graph them and count rise over run.