Answer :
Sure! Let's solve this step-by-step:
### Part (a): Graph the line through the points (-6, 4) and (6, -4).
1. Identify the points: We have the points [tex]\((-6, 4)\)[/tex] and [tex]\((6, -4)\)[/tex].
2. Plot the points:
- Point [tex]\((-6, 4)\)[/tex] is located 6 units to the left of the origin and 4 units up.
- Point [tex]\((6, -4)\)[/tex] is located 6 units to the right of the origin and 4 units down.
3. Draw the line:
- Draw a straight line passing through both points. This line represents the linear relationship between [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that goes through these points.
### Part (b): Write an equation of the line in the form [tex]\( y = mx + b \)[/tex].
1. Calculate the slope [tex]\( m \)[/tex]:
The slope [tex]\( m \)[/tex] of a line passing through the points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
For our points [tex]\((x_1, y_1) = (-6, 4)\)[/tex] and [tex]\((x_2, y_2) = (6, -4)\)[/tex]:
[tex]\[ m = \frac{-4 - 4}{6 - (-6)} = \frac{-8}{6 + 6} = \frac{-8}{12} = -\frac{2}{3} \][/tex]
2. Find the y-intercept [tex]\( b \)[/tex]:
The equation of the line is [tex]\( y = mx + b \)[/tex]. To find [tex]\( b \)[/tex], use one of the points. We'll use [tex]\((-6, 4)\)[/tex]:
[tex]\[ y = mx + b \\ 4 = \left(-\frac{2}{3}\right)(-6) + b \\ 4 = 4 + b \\ b = 4 - 4 \\ b = 0 \][/tex]
3. Write the equation:
The equation of the line in slope-intercept form [tex]\( y = mx + b \)[/tex] is:
[tex]\[ y = -\frac{2}{3}x + 0 \\ y = -\frac{2}{3}x \][/tex]
So, the line through the points [tex]\((-6, 4)\)[/tex] and [tex]\((6, -4)\)[/tex] has the equation [tex]\( y = -\frac{2}{3}x \)[/tex].
### Part (a): Graph the line through the points (-6, 4) and (6, -4).
1. Identify the points: We have the points [tex]\((-6, 4)\)[/tex] and [tex]\((6, -4)\)[/tex].
2. Plot the points:
- Point [tex]\((-6, 4)\)[/tex] is located 6 units to the left of the origin and 4 units up.
- Point [tex]\((6, -4)\)[/tex] is located 6 units to the right of the origin and 4 units down.
3. Draw the line:
- Draw a straight line passing through both points. This line represents the linear relationship between [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that goes through these points.
### Part (b): Write an equation of the line in the form [tex]\( y = mx + b \)[/tex].
1. Calculate the slope [tex]\( m \)[/tex]:
The slope [tex]\( m \)[/tex] of a line passing through the points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
For our points [tex]\((x_1, y_1) = (-6, 4)\)[/tex] and [tex]\((x_2, y_2) = (6, -4)\)[/tex]:
[tex]\[ m = \frac{-4 - 4}{6 - (-6)} = \frac{-8}{6 + 6} = \frac{-8}{12} = -\frac{2}{3} \][/tex]
2. Find the y-intercept [tex]\( b \)[/tex]:
The equation of the line is [tex]\( y = mx + b \)[/tex]. To find [tex]\( b \)[/tex], use one of the points. We'll use [tex]\((-6, 4)\)[/tex]:
[tex]\[ y = mx + b \\ 4 = \left(-\frac{2}{3}\right)(-6) + b \\ 4 = 4 + b \\ b = 4 - 4 \\ b = 0 \][/tex]
3. Write the equation:
The equation of the line in slope-intercept form [tex]\( y = mx + b \)[/tex] is:
[tex]\[ y = -\frac{2}{3}x + 0 \\ y = -\frac{2}{3}x \][/tex]
So, the line through the points [tex]\((-6, 4)\)[/tex] and [tex]\((6, -4)\)[/tex] has the equation [tex]\( y = -\frac{2}{3}x \)[/tex].
