Problem 58
Question
Find the \(x\) - and \(y\) -intercepts for each line and use them to graph the line. $$y=\frac{1}{3} x+10$$
Step-by-Step Solution
Verified Answer
The y-intercept is (0, 10) and the x-intercept is (-30, 0).
1Step 1: Identify the y-intercept
The y-intercept is found by setting \(x = 0\) in the equation. The equation becomes: \[ y = \frac{1}{3} \times 0 + 10 = 10 \]So, the y-intercept is \((0, 10)\).
2Step 2: Identify the x-intercept
The x-intercept is found by setting \(y = 0\) in the equation. Start by solving for \(x\): \[ 0 = \frac{1}{3} x + 10 \]Subtract 10 from both sides: \[ -10 = \frac{1}{3} x \]Now, multiply both sides by 3: \[ x = -30 \]So, the x-intercept is \((-30, 0)\).
3Step 3: Graph the line using intercepts
Plot the intercepts \((0, 10)\) and \((-30, 0)\) on the coordinate plane. Draw a straight line through these points to represent the equation \(y = \frac{1}{3} x + 10\).
Key Concepts
x-intercepty-interceptlinear equationsgraphing lines
x-intercept
The x-intercept of a line is the point where the graph crosses the x-axis. This means that at the x-intercept, the value of y is always zero. To find the x-intercept, you need to set the y value to zero in the given linear equation and solve for x. For example, in the equation \(y = \frac{1}{3} x + 10\), to find the x-intercept, set \(y = 0\) and solve for x:
\[ 0 = \frac{1}{3} x + 10 \]
Subtract 10 from both sides:
\[ -10 = \frac{1}{3} x \]
Now, multiply both sides by 3:
\[ x = -30 \]
This means the x-intercept is at the point \((-30, 0)\). This point is where the line crosses the x-axis.
\[ 0 = \frac{1}{3} x + 10 \]
Subtract 10 from both sides:
\[ -10 = \frac{1}{3} x \]
Now, multiply both sides by 3:
\[ x = -30 \]
This means the x-intercept is at the point \((-30, 0)\). This point is where the line crosses the x-axis.
y-intercept
The y-intercept of a line is the point where the graph crosses the y-axis. At the y-intercept, the value of x is always zero. To find the y-intercept, set the x value to zero in the given linear equation and solve for y. For example, in the equation \(y = \frac{1}{3} x + 10\), to find the y-intercept, set \(x = 0\) and solve for y:
\[ y = \frac{1}{3} \times 0 + 10 \]
Since \(\frac{1}{3} \times 0 = 0\), we have
\[ y = 10 \]
This means the y-intercept is at the point \((0, 10)\). This point is where the line crosses the y-axis.
\[ y = \frac{1}{3} \times 0 + 10 \]
Since \(\frac{1}{3} \times 0 = 0\), we have
\[ y = 10 \]
This means the y-intercept is at the point \((0, 10)\). This point is where the line crosses the y-axis.
linear equations
Linear equations describe straight lines on a graph. A linear equation is typically written in the form \(y = mx + b\), where m is the slope of the line, and b is the y-intercept. The slope indicates how steep the line is, and the y-intercept indicates where the line crosses the y-axis. For example, in the equation \(y = \frac{1}{3} x + 10\), the slope (m) is \(\frac{1}{3}\), and the y-intercept (b) is 10. This tells us that for every unit increase in x, y increases by \(\frac{1}{3}\). Linear equations can be rearranged and manipulated to find specific points such as x-intercepts and y-intercepts.
graphing lines
Graphing lines involves plotting points on a coordinate plane and drawing a straight line through them. To graph a line from a linear equation, first identify the intercepts. Plot the x-intercept and y-intercept on the coordinate plane. For example, using the equation \(y = \frac{1}{3} x + 10\), we found the intercepts to be \((-30, 0)\) and \((0, 10)\). Next, plot these points:
- x-intercept: \((-30, 0)\)
- y-intercept: \((0, 10)\)
Other exercises in this chapter
Problem 57
Find the \(x\) - and \(y\) -intercepts for each line and use them to graph the line. $$y=-\frac{1}{2} x-20$$
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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
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Determine whether the lines \(l_{1}\) and \(l_{2}\) are parallel, perpendicular, or neither. \(l_{1}\) goes through \((0,3)\) and \((4,17), l_{2}\) goes through
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Determine whether the lines \(l_{1}\) and \(l_{2}\) are parallel, perpendicular, or neither. \(l_{1}\) goes through \((-2,-3)\) and \((4,1), l_{2}\) goes throug
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