Problem 57
Question
Find the \(x\) - and \(y\) -intercepts for each line and use them to graph the line. $$y=-\frac{1}{2} x-20$$
Step-by-Step Solution
Verified Answer
The y-intercept is (0, -20) and the x-intercept is (-40, 0). Plot these points and draw the line.
1Step 1: Find the y-intercept
The y-intercept is found by setting x = 0 in the equation. Substitute x = 0 into the equation: \[ y = -\frac{1}{2} \times 0 - 20 = -20 \] Therefore, the y-intercept is (0, -20).
2Step 2: Find the x-intercept
The x-intercept is found by setting y = 0 in the equation. Substitute y = 0 into the equation and solve for x: \[ 0 = -\frac{1}{2} x - 20 \] Rearrange and solve for x: \[ \frac{1}{2} x = -20 \] \[ x = -20 \times 2 = -40 \] Therefore, the x-intercept is (-40, 0).
3Step 3: Graph the line
To graph the line, plot the intercepts found in the previous steps: (0, -20) and (-40, 0). Draw a line through these points to represent the equation \( y = -\frac{1}{2} x - 20 \).
Key Concepts
x-intercepty-interceptslope-intercept form
x-intercept
In a linear equation, the x-intercept is where the line crosses the x-axis. This happens when y equals zero. To find the x-intercept, substitute y = 0 into the equation and solve for x.
For the given equation: \[ y = -\frac{1}{2} x - 20 \] Set y = 0: \[ 0 = -\frac{1}{2} x - 20 \] This means: \[ \frac{1}{2} x = -20 \] To isolate x, multiply both sides by 2: \[ x = -40 \] So, the x-intercept is at the point (-40, 0).
When you graph it, you will place a point on the x-axis at -40. This is where the line will cross horizontally.
For the given equation: \[ y = -\frac{1}{2} x - 20 \] Set y = 0: \[ 0 = -\frac{1}{2} x - 20 \] This means: \[ \frac{1}{2} x = -20 \] To isolate x, multiply both sides by 2: \[ x = -40 \] So, the x-intercept is at the point (-40, 0).
When you graph it, you will place a point on the x-axis at -40. This is where the line will cross horizontally.
y-intercept
The y-intercept of a line is where the line crosses the y-axis. This happens when x equals zero. To find the y-intercept, substitute x = 0 into the equation and solve for y.
For the given equation: \[ y = -\frac{1}{2} x - 20 \] Set x = 0: \[ y = -\frac{1}{2} \times 0 - 20 = -20 \] So, the y-intercept is at the point (0, -20).
When you graph it, you will place a point on the y-axis at -20. This is where the line will cross vertically.
For the given equation: \[ y = -\frac{1}{2} x - 20 \] Set x = 0: \[ y = -\frac{1}{2} \times 0 - 20 = -20 \] So, the y-intercept is at the point (0, -20).
When you graph it, you will place a point on the y-axis at -20. This is where the line will cross vertically.
slope-intercept form
The slope-intercept form of a linear equation is written as: \[ y = mx + b \] Where:
- \
Other exercises in this chapter
Problem 56
Find the \(x\) - and \(y\) -intercepts for each line and use them to graph the line. $$y=-3 x+6$$
View solution Problem 56
Determine whether the lines \(l_{1}\) and \(l_{2}\) are parallel, perpendicular, or neither. \(l_{1}\) goes through \((4,3)\) and \((2,6), l_{2}\) goes through
View solution Problem 57
Determine whether the lines \(l_{1}\) and \(l_{2}\) are parallel, perpendicular, or neither. \(l_{1}\) goes through \((0,0)\) and \((-2,5), l_{2}\) goes through
View solution Problem 58
Find the \(x\) - and \(y\) -intercepts for each line and use them to graph the line. $$y=\frac{1}{3} x+10$$
View solution