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Chapter 5: Problem 4

b. Find the equation of the line through each pair of points. $$ (1,1) \text { and }(3,0) $$

Short Answer

Expert verified
y = -\frac{1}{2}x + \frac{3}{2}

Step by step solution

01

- Find the Slope

Use the formula for the slope of a line passing through two points \(x_1, y_1\) and \(x_2, y_2\): \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Given the points (1,1) and (3,0), substitute the values into the formula: \[ m = \frac{0 - 1}{3 - 1} = \frac{-1}{2} = -\frac{1}{2} \]
02

- Use Point-Slope Form

The point-slope form of a line's equation is given by \[ y - y_1 = m(x - x_1) \] Using the slope \(m = -\frac{1}{2}\) and one of the points, say (1,1), substitute them into the point-slope form equation: \[ y - 1 = -\frac{1}{2}(x - 1) \]
03

- Simplify to Slope-Intercept Form

Simplify the equation from Step 2 to put it in slope-intercept form \(y = mx + b\): \[ y - 1 = -\frac{1}{2}(x - 1) \] Distribute the slope: \[ y - 1 = -\frac{1}{2}x + \frac{1}{2} \] Add 1 to both sides to isolate \(y\): \[ y = -\frac{1}{2}x + \frac{1}{2} + 1 \] Combine like terms: \[ y = -\frac{1}{2}x + \frac{3}{2} \]

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Key Concepts

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

Slope Calculation
To find the equation of a line, the first step is to calculate the slope. The slope is a measure of how steep the line is. It's calculated using two points on the line. The formula to find the slope (or gradient) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] \For instance, consider the points \( (1, 1) \) and \( (3, 0) \). Substituting these into the formula: \[ m = \frac{0 - 1}{3-1} = \frac{-1}{2} \] When you plug in the values, you'll find the slope is \( -\frac{1}{2} \). The negative sign indicates the line slopes downwards from left to right.
Point-Slope Form
After finding the slope, you can use the point-slope form to write the equation of the line. The point-slope form is useful if you know one point on the line and the slope. The formula is: \[ y - y_1 = m(x - x_1) \] \Now use our slope \( -\frac{1}{2} \) and one of the points, say \( (1, 1) \). Substitute these values into the point-slope form: \[ y - 1 = -\frac{1}{2} (x - 1) \] This form of the equation is already useful, especially when dealing with problems involving finding specific points on the line. It shows a clear relationship between the x and y coordinates based on the slope.
Slope-Intercept Form
To make the equation more familiar and easy to work with, you can simplify the point-slope form into the slope-intercept form \( y = mx + b \). This form clearly shows the y-intercept (where the line crosses the y-axis) and the slope. Here鈥檚 how to convert our point-slope form to slope-intercept form: \[ y - 1 = -\frac{1}{2} (x - 1) \] \First, distribute the slope: \[ y - 1 = -\frac{1}{2}x + \frac{1}{2} \] \Next, isolate y by adding 1 to both sides: \[ y = -\frac{1}{2}x + \frac{1}{2} + 1 \] \Combine like terms to get: \[ y = -\frac{1}{2}x + \frac{3}{2} \] \Now, you have the equation in slope-intercept form, which makes it very easy to plot the line on a graph.

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Most popular questions from this chapter

For each of the following, graph the given points. $$ (1,4) $$

b. Find the equation of the line through each pair of points. $$ (-1,6) \text { and }(5,-2) $$

For each of the following, graph the given points. $$ (0,0) $$

Change each of the following equations to slope-intercept form, and find the slope. $$ 5 x-y+2=0 $$

For each of the following, graph the given points. $$ (3,-1) $$
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