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Chapter 3: Problem 69

Each table of values gives several points that lie on a line. (a) Use any two of the ordered pairs to find the slope of the line. (b) Identify the y-intercept of the line. (c) Use the slope and y-intercept from parts (a) and (b) to write an equation of the line in slope-intercept form. (d) Graph the equation. $$ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & -1 \\ \hline 3 & 5 \\ \hline 5 & 9 \end{array} $$

Short Answer

Expert verified
The equation of the line is \[ y = 2x - 1 \].

Step by step solution

01

Choose two points

Select any two points from the table of values. Let鈥檚 choose the points (0, -1) and (3, 5).
02

Calculate the slope

Utilize the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \].Plug in the coordinates of the chosen points: \[ m = \frac{5 - (-1)}{3 - 0} = \frac{6}{3} = 2 \].So, the slope (m) is 2.
03

Identify the y-intercept

The y-intercept is the value of y when x equals 0. From the table, when x is 0, y is -1. Thus, the y-intercept (b) is -1.
04

Write the equation in slope-intercept form

Use the slope-intercept form equation \[ y = mx + b \].Substitute the slope (m) and y-intercept (b): \[ y = 2x - 1 \].Thus, the equation of the line is \[ y = 2x - 1 \].
05

Graph the equation

To graph the equation \[ y = 2x - 1 \], start with the y-intercept (0, -1). From there, use the slope to find the next point. The slope is 2, which means rise over run = 2 over 1. From (0, -1), move up 2 units and right 1 unit to reach (1, 1). Draw a line through these points and extend it in both directions.

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Key Concepts

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

Linear Equations
Linear equations describe straight lines on a graph. Typically, a linear equation is written in the form of \(y = mx + b\). Here:
  • \(m\) represents the slope of the line.
  • \(b\) represents the y-intercept, or where the line crosses the y-axis.
Linear equations make it easier to predict values, create graphs, and understand relationships between variables. By identifying the slope and y-intercept, we unlock the power to visualize data through graphing.
Slope
The slope of a line shows how steep the line is and the direction it鈥檚 going. We calculate the slope (\(m\)) using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \] This formula represents the change in \(y\) over the change in \(x\) (also known as 'rise over run'). Let's see an example: If we choose points (0, -1) and (3, 5) from the table, we plug them into the formula: \[ m = \frac{5 - (-1)}{3 - 0} = \frac{6}{3} = 2 \] So, the slope (\(m\)) is 2, indicating that for every 1 unit we move right on the x-axis, we move up 2 units on the y-axis.
Y-Intercept
The y-intercept (\(b\)) of a line is the y-value where the line crosses the y-axis (where \(x = 0\)). We can find this directly from our table:
From the given values, when \(x = 0\), \(y = -1\).
Thus, the y-intercept (\(b\)) is -1. This point is crucial because it anchors the line on the graph, helping us visualize where it starts or intersects the y-axis.
Graphing
To graph a linear equation, like \(y = 2x - 1\), follow these steps:
  • Start with the y-intercept. For this equation, it鈥檚 (0, -1). Plot this point on the graph.
  • Use the slope to find the next point. Our slope is 2, meaning 鈥榬ise over run鈥� is 2/1. From (0, -1), move up 2 units and right 1 unit to reach (1, 1) and place a point there.
  • Draw a line through these points and extend it in both directions. This line is the visual representation of our linear equation.
By following these steps, you can turn an equation into a clear, visual graph. Graphing helps in understanding the relationship between x and y values and allows us to make predictions.

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Most popular questions from this chapter

Find the \(x\) - and \(y\) -intercepts for the graph of each equation. $$ 4 x-5 y=0 $$

What is the slope of a line whose graph is (a) parallel to the graph of \(3 x+y=7 ?\) (b) perpendicular to the graph of \(3 x+y=7 ?\)

Write an equation of the line satisfying the given conditions. Give the final answer in slope-intercept form. (Hint: Recall the relationships among slopes of parallel and perpendicular lines.) Parallel to \(5 x-y=10\) \(y\) -intercept (0,-2)

Each table of values gives several points that lie on a line. (a) Use any two of the ordered pairs to find the slope of the line. (b) What is the x-intercept of the line? The y-intercept? (c) Graph the line. $$ \begin{array}{r|r} \hline x & y \\ \hline-4 & 0 \\ \hline-2 & 2 \\ \hline 0 & 4 \\ \hline 1 & 5 \end{array} $$

Graph each linear equation. $$ x=-1 $$
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