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Chapter 1: Problem 64

Find a linear equation whose graph is the straight line with the given properties. Through \((1,-4)\) and \((-1,-1)\)

Short Answer

Expert verified
The equation of the straight line that passes through the given points \((1, -4)\) and \((-1, -1)\) is \(y = -\frac{3}{2}x -\frac{5}{2}\).

Step by step solution

01

Find the slope (m) between the given points

To find the slope of the line passing through the given points \((1, -4)\) and \((-1, -1)\), we can use the formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\) Here, \((x_1, y_1) = (1, -4)\) and \((x_2, y_2) = (-1, -1)\). Now, substitute the values and solve for the value of \(m\): \(m = \frac{-1 - (-4)}{-1 - 1} = \frac{3}{-2} = -\frac{3}{2}\) The slope of the line is \(-\frac{3}{2}\).
02

Find the y-intercept (b) using slope-intercept form

Now it's time to find the y-intercept of the line. We can use either of the given points for this step. Let's use the point \((1, -4)\). The slope-intercept form of the line is given by: \(y = mx + b\) Replace \(x\), \(y\), and \(m\) with the values we have found so far: \(-4 = -\frac{3}{2}(1) + b\) Now solve for \(b\): \(b = -4 + \frac{3}{2} = -\frac{5}{2}\) The y-intercept of the line is \(-\frac{5}{2}\).
03

Write the equation of the line

Finally, we have the slope and y-intercept of the line. Plug these values into the slope-intercept equation \(y = mx + b\): \(y = -\frac{3}{2}x -\frac{5}{2}\) So, the equation of the straight line that passes through the given points \((1, -4)\) and \((-1, -1)\) is: \(y = -\frac{3}{2}x -\frac{5}{2}\)

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Key Concepts

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

Slope of a Line
Understanding the slope of a line is foundational in algebra and helps us to grasp how steep a line is when graphed. The slope is represented by the letter m and is calculated as the rise over the run between two points on the line. In other words, it's the change in the y-coordinate (vertical change) divided by the change in the x-coordinate (horizontal change).

The formula to find the slope m between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
For example, with the points \( (1, -4) \) and \( (-1, -1) \), we get a slope of \( -\frac{3}{2} \). This negative sign shows that the line is inclining downwards as it moves from left to right.
Y-intercept
Every linear equation graph crosses the y-axis at a certain point called the y-intercept, denoted as \( b \). It indicates the value of \( y \) when \( x = 0 \). To find the y-intercept using a point \((x, y)\) on the line and the slope \(m\), we substitute these values into the slope-intercept form \(y = mx + b\) and solve for \(b\).

In our example, substituting \( -4 \) for \(y\), \( -\frac{3}{2} \) for \(m\), and \(1\) for \(x\), we can solve for \(b\):
\[ b = -4 + \frac{3}{2} \]
This calculation reveals that the y-intercept is \(-\frac{5}{2}\). This is the point where the line will cross the y-axis, which is crucial for graphing the linear equation.
Slope-Intercept Form
The slope-intercept form is a way of writing a linear equation that makes it particularly easy to graph. This form is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. It clearly shows the steepness of the line and where it intersects the y-axis.

Manipulating this formula allows us to isolate \(y\) and reveal the direction and initial value of the line. For instance, in our exercise, the final linear equation becomes: \[ y = -\frac{3}{2}x -\frac{5}{2} \]. The negative slope indicates a downward direction, and the y-intercept tells us that the line starts at the point \((0, -\frac{5}{2})\) on the y-axis.
Graphing Linear Equations
When it comes to graphing linear equations, knowing the slope and y-intercept is immensely helpful. To plot the line, follow these steps: Start by plotting the y-intercept on the vertical axis. Using our example, you would place a point at \((0, -\frac{5}{2})\). Then, use the slope to determine the rise over run from that point. With a slope of \( -\frac{3}{2} \), we'd move down 3 units and to the right 2 units to plot a second point.

Once you have at least two points marked on the graph, simply connect them with a straight line, extending it in both directions. Always remember that a linear equation will produce a straight line, and every point on that line is a solution to the equation.

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Most popular questions from this chapter

If \(y\) and \(x\) are related by the linear expression \(y=m x+b\), how will \(y\) change as \(x\) changes if \(m\) is positive? negative? zero?

The number of Sirius Satellite Radio subscribers grew from \(0.3\) million in 2003 to \(3.2\) million in \(2005 .^{32}\) a. Use this information to find a linear model for the number \(N\) of subscribers (in millions) as a function of time \(t\) in years since 2000 . b. Give the units of measurement and interpretation of the slope. c. Use the model from part (a) to predict the 2006 figure. (The actual 2006 figure was approximately 6 million subscribers.)

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