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Chapter 1: Problem 17

Find an equation of the line passing through the given points. $$ (3,4) \text { and }(7,8) $$

Short Answer

Expert verified
The equation of the line is \(y = x + 1\).

Step by step solution

01

- Calculate the slope

The first step is to calculate the slope of the line passing through the points \(3, 4\) and \(7, 8\). The slope \(m\) is given by the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting the points into the formula, we get: \[ m = \frac{8 - 4}{7 - 3} = \frac{4}{4} = 1 \]
02

- Use the point-slope formula

Next, use the point-slope formula to find the equation of the line. The point-slope form of a line is given by: \[ y - y_1 = m(x - x_1) \] Using the slope from Step 1 and one of the points \(3, 4\), we get: \[ y - 4 = 1(x - 3) \]
03

- Simplify to slope-intercept form

Finally, simplify the equation to the slope-intercept form \(y = mx + b\): \[ y - 4 = x - 3 \] Adding 4 to both sides, we get: \[ y = x + 1 \]

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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, we start by calculating the slope. The slope, often represented as \(m\), is a measure of how steep the line is. It shows how much the y-coordinate of a point on the line changes for a unit change in the x-coordinate. The formula for calculating the slope is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Here, \((x_1, y_1)\) and \((x_2, y_2)\) are two distinct points on the line.
For the points (3, 4) and (7, 8), we substitute them into the formula as follows:
\[ m = \frac{8 - 4}{7 - 3} = \frac{4}{4} = 1 \]
Hence, the slope of the line passing through these points is 1.
point-slope formula
Once we have the slope, we use the point-slope formula to find the equation of the line. The point-slope form is particularly useful when we have the slope of the line and a point it passes through. The formula is:
\[ y - y_1 = m(x - x_1) \]
Plugging in the values from our example, where the slope \(m\) is 1 and the point is (3, 4), we get:
\[ y - 4 = 1(x - 3) \]
This equation now represents the line in point-slope form. It shows the relationship between any point \((x, y)\) on the line with the point (3, 4).
slope-intercept form
To make it easier to read and use for plotting, we convert the equation to the slope-intercept form, which is \(y = mx + b\).
The slope-intercept form clearly shows the slope \(m\) and the y-intercept \(b\), which is where the line crosses the y-axis. From the point-slope form:
\[ y - 4 = x - 3 \]
We simplify this by adding 4 to both sides to solve for \(y\):
\[ y = x + 1 \]
Now, the equation is in slope-intercept form with a slope \(m\) of 1 and a y-intercept \(b\) of 1. So, the line has the equation \(y = x + 1\). This is very useful for graphing and understanding the line's characteristics.

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Most popular questions from this chapter

Each point below satisfies one of the equations. Match each equation with a point. $$\left(\begin{array}{ccc}{-1,1} & {(7,0)} & {(-1,-1)} & {(10,1)} & {(-2,0)} & {(-1,0.9)}\end{array}\right.$$ a. \(y=2 x-14\) b. \(y=x^{2}-4\) c. \(y=0.1 x+x^{2}\) d. \(y=x^{3}\) e. \(y=-x^{3}\) f. \(y=x^{2}-99\)

For Exercises 20鈥�28, answer Parts a and b. a. What is the constant difference between the \(y\) values as the \(x\) values increase by 1\(?\) b. What is the constant difference between the \(y\) values as the \(x\) values decrease by 2\(?\) $$ y=-2 x+12 $$

Each table describes a linear relationship. For each relationship, find the slope of the line and the \(y\) -intercept. Then write an equation for the relationship in the form \(y=m x+b .\) $$\begin{array}{|c|c|c|c|c|c|}\hline x & {-8} & {-3} & {3} & {5} & {10} \\\ \hline y & {26} & {11} & {-7} & {-13} & {-28} \\ \hline\end{array}$$

The table shows x and y values for a particular relationship. $$\begin{array}{|c|c|c|c|c|}\hline x & {6} & {3} & {1} & {2.5} \\ \hline y & {7} & {1} & {-3} & {0} \\ \hline\end{array}$$ $$\begin{array}{l}{\text { a. Graph the ordered pairs }(x, y) . \text { Make each axis scale from }-10} \\ {\text { to } 10 .} \\ {\text { b. Could the points represent a linear relationship? If so, write an }} \\ {\text { equation for the line. }} \\ {\text { c. From your graph, predict the } y \text { value for an } x \text { value of }-2 . \text { Check }} \\ {\text { your answer by substituting it into the equation. }}\end{array}$$ $$ \begin{array}{l}{\text { d. From your graph, find the } x \text { value for a } y \text { value of }-2 . \text { Check your }} \\ {\text { answer by substituting it into the equation. }} \\ {\text { e. Use your equation to find the } y \text { value for each of these } x \text { values: }} \\ {0,-- 1,-1.5,-2.5 . \text { Check that the corresponding points all lie on }} \\\ {\text { the line. }}\end{array} $$

Evaluate each expression for \(a=2\) and \(b=3\). $$ \left(\frac{a}{b}+a\right)^{a} $$
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