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Chapter 4: Problem 35

For the following exercises, given each set of information, find a linear equation satisfying the conditions, if possible. \(x\) intercept at (-2,0) and \(y\) intercept at (0,-3)

Short Answer

Expert verified
The equation is \(y = \frac{-3}{2}x - 3\).

Step by step solution

01

Understand the Intercepts

The problem provides two points: the \(x\)-intercept at \((-2,0)\) and the \(y\)-intercept at \((0,-3)\). The \(x\)-intercept means where the line crosses the x-axis, and the \(y\)-intercept meets the y-axis.
02

Use Intercepts to Determine the Slope

The slope \(m\) of the line can be calculated using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). We use the points \((-2,0)\) and \((0,-3)\). So, \(m = \frac{-3 - 0}{0 + 2} = \frac{-3}{2}\).
03

Write the Equation in Point-Slope Form

Using the point-slope form \(y - y_1 = m(x - x_1)\) with the slope \(m = \frac{-3}{2}\) and one of the points, \(x_1 = 0\) and \(y_1 = -3\), substitute: \[y + 3 = \frac{-3}{2}(x - 0)\].
04

Simplify to Slope-Intercept Form

Simplify the equation from point-slope form to slope-intercept form \(y = mx + b\): \[y + 3 = \frac{-3}{2}x\]. Subtract 3 from both sides to write: \[y = \frac{-3}{2}x - 3\].
05

Verify the Equation

Check that the line satisfies both intercepts. Substitute \(x = -2\) in \(y = \frac{-3}{2}x - 3\) to confirm \(y = 0\), and substitute \(y = -3\) to confirm \(x = 0\). Both verifications are correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

X-Intercept
The x-intercept of a line is a foundational concept in linear equations. It represents the point where the line crosses the x-axis. This means the value of y at the x-intercept is always zero.
Understanding x-intercepts is key to graphing and interpreting linear equations. In the equation context from our exercise, the x-intercept is at (-2, 0). At this point, the line meets the x-axis. This tells us a specific detail about the positioning and angle of the line.
  • To find the x-intercept from a linear equation, set y to zero and solve for x.
  • Remember, the x-coordinate of the x-intercept does not give the line's slope or steepness, but it does indicate one of the endpoints for slope calculation.
Knowing this intercept helps us establish the structure of the line when combined with another point.
Y-Intercept
The y-intercept is equally important. It is the point where the line crosses the y-axis. At this intercept, the value of x is always zero.
This detail often gives a direct insight into the starting value of a linear pattern when x is initially zero. In our example, the y-intercept is located at (0, -3). This means when the line crosses the y-axis, it does so at y = -3.
  • To locate the y-intercept from a linear equation like the one in slope-intercept form, simply find the constant term (b).
  • This intercept is a crucial reference point for graphing the line.
Recognizing the y-intercept is essential for fully graphing and understanding the behavior of linear equations.
Point-Slope Form
The point-slope form of a linear equation is a convenient way to write the equation when you know the slope and a single point on the line. This form is: \[y - y_1 = m(x - x_1)\]where \(m\) is the slope and \((x_1, y_1)\) is the known point.

This form is incredibly useful because:
  • You can easily create the equation from just one point and the slope.
  • It is a stepping stone to other forms, like the slope-intercept form.
In our case, using the slope \(\frac{-3}{2}\) and y-intercept point \((0, -3)\), the point-slope form is \[y + 3 = \frac{-3}{2}(x - 0)\]. This expression effectively describes our linear equation as it includes our key linear elements: slope and intercept.
Slope Calculation
Calculating the slope of a line is vital in understanding how steeply the line inclines or declines. The slope (\(m\)) is determined using the formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where \((x_1, y_1)\) and \((x_2, y_2)\) are any two points on the line.
For our problem, using the intercept points \((-2, 0)\) and \((0, -3)\), the slope is \(\frac{-3 - 0}{0 + 2} = \frac{-3}{2}\). This negative slope indicates that our line moves downward as it progresses from left to right.
  • The slope tells us how much y changes for a one-unit change in x.
  • A negative slope like ours means the line is decreasing, reflecting in the linear equation as the negative coefficient of x.
Remember, mastering slope calculation is crucial for constructing, interpreting, and graphing linear equations.

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Most popular questions from this chapter

For the following exercises, sketch the graph of each equation. $$ f(x)=-3 x+2 $$

For the following exercises, sketch a line with the given features. A \(y\) -intercept of (0,3) and slope \(\frac{2}{5}\)

For the following exercises, draw a scatter plot for the data provided. Does the data appear to be linearly related? $$ \begin{array}{|l|l|l|l|l|l|} \hline 1 & 3 & 5 & 7 & 9 & 11 \\ \hline 1 & 9 & 28 & 65 & 125 & 216 \\ \hline \end{array} $$

For the following exercises, write an equation for the line described. Write an equation for a line parallel to \(g(x)=3 x-1\) and passing through the point (4,9) .

For the following exercises, given each set of information, find a linear equation satisfying the conditions, if possible. \(f(-1)=4,\) and \(f(5)=1\)
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