Question

a line passes through the points (-4, -2) and (6, 3). write its equation in slope-intercept form. write your answer using integers, proper fractions, and improper fractions in simplest form.

Explanation:

Step1: Calculate the slope

The slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Here, \( (x_1,y_1)=(-4,-2) \) and \( (x_2,y_2)=(6,3) \). So, \( m=\frac{3 - (-2)}{6 - (-4)}=\frac{3 + 2}{6 + 4}=\frac{5}{10}=\frac{1}{2} \).

Step2: Use point - slope form to find the equation

The point - slope form is \( y - y_1=m(x - x_1) \). Using the point \( (-4,-2) \) and \( m = \frac{1}{2} \), we have \( y-(-2)=\frac{1}{2}(x - (-4)) \), which simplifies to \( y + 2=\frac{1}{2}(x + 4) \).

Step3: Convert to slope - intercept form (\( y=mx + b \))

Expand the right - hand side: \( y+2=\frac{1}{2}x+2 \). Subtract 2 from both sides: \( y=\frac{1}{2}x+2 - 2 \), so \( y=\frac{1}{2}x \). Wait, that's wrong. Wait, let's recalculate step 2. Using the point \( (6,3) \) and \( m=\frac{1}{2} \), the point - slope form is \( y - 3=\frac{1}{2}(x - 6) \). Expand: \( y-3=\frac{1}{2}x-3 \). Add 3 to both sides: \( y=\frac{1}{2}x \). Wait, but when we use the first point \( (-4,-2) \): \( y+2=\frac{1}{2}(x + 4)\Rightarrow y+2=\frac{1}{2}x + 2\Rightarrow y=\frac{1}{2}x \). Wait, but let's check with the two points. For \( x=-4 \), \( y=\frac{1}{2}\times(-4)=-2 \), which matches. For \( x = 6 \), \( y=\frac{1}{2}\times6 = 3 \), which also matches. So the equation is \( y=\frac{1}{2}x \). Wait, no, wait, my mistake in the first calculation of the slope? Wait, \( (y_2 - y_1)=3-(-2)=5 \), \( (x_2 - x_1)=6-(-4)=10 \), so slope is \( \frac{5}{10}=\frac{1}{2} \), that's correct. Then using point - slope: \( y - y_1=m(x - x_1) \). Let's use \( (6,3) \): \( y - 3=\frac{1}{2}(x - 6) \), \( y-3=\frac{1}{2}x-3 \), \( y=\frac{1}{2}x \). Yes, that's correct.

Wait, but let's do it again. The slope formula: \( m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{3-(-2)}{6-(-4)}=\frac{5}{10}=\frac{1}{2} \). Then, using the slope - intercept form \( y=mx + b \), we can plug in one of the points to find \( b \). Let's use \( (6,3) \): \( 3=\frac{1}{2}\times6 + b \), \( 3 = 3 + b \), so \( b = 0 \). So the equation is \( y=\frac{1}{2}x \).

Answer:

\( y=\frac{1}{2}x \)