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Chapter 15: Problem 50

Find the slope-intercept equation of a line given the conditions. The graph contains the points \((5,1)\) and \((8,0)\)

Short Answer

Expert verified
y = -\frac{1}{3}x + \frac{8}{3}

Step by step solution

01

- Find the Slope

The slope of a line passing through two points \(x_1, y_1\) and \(x_2, y_2\) can be found using the formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substitute \(x_1 = 5, y_1 = 1, x_2 = 8, y_2 = 0\). Thus, the slope \(m \) is: \[m = \frac{0 - 1}{8 - 5} = \frac{-1}{3}\]
02

- Find the Y-Intercept

Use the point-slope form of the equation \(y - y_1 = m(x - x_1)\) to find the y-intercept (b). Substitute \(m = -\frac{1}{3}\) and one of the given points, say \( (5, 1) \). \[ y - 1 = -\frac{1}{3}(x - 5) \]
03

- Solve for y

Convert the equation into slope-intercept form \(y = mx + b\). Simplify the point-slope form equation from Step 2: \[ y - 1 = -\frac{1}{3}(x - 5) \] \[ y - 1 = -\frac{1}{3}x + \frac{5}{3} \] \[ y = -\frac{1}{3}x + \frac{5}{3} + 1 \] \[ y = -\frac{1}{3}x + \frac{8}{3} \]

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Key Concepts

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

slope of a line
The slope of a line measures its steepness and direction. To find the slope that passes through two points \(x_1, y_1\) and \(x_2, y_2\), you use the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Here, 'm' represents the slope.

Let's apply this formula to the points \(5, 1\) and \(8, 0\). Substitute the values to get: \[ m = \frac{0 - 1}{8 - 5} = \frac{-1}{3} \].

  • The numerator (difference in y-values) shows the vertical change.
  • The denominator (difference in x-values) reflects the horizontal change.

The slope \(-\frac{1}{3}\) indicates the line decreases by 1 unit vertically for every 3 units it moves horizontally.
point-slope form
Point-slope form is an equation of a line that provides the slope and a point on the line. The equation is given as: \(y - y_1 = m(x - x_1)\).

Here, \(m\) is the slope, and \(x_1, y_1\) is a specific point on the line. Using the slope \(-\frac{1}{3}\) and the point \( (5, 1) \), we substitute the values in: \[ y - 1 = -\frac{1}{3}(x - 5) \].

  • This form is helpful to quickly identify the slope and a specific point on the line.
  • It's often used as an intermediate step to find the slope-intercept form.
finding y-intercept
Finding the y-intercept means calculating the value of 'b' in the slope-intercept form \(y = mx + b\). This is the point where the line crosses the y-axis (where \(x = 0\)).

Starting with the point-slope form \[y - 1 = -\frac{1}{3}(x - 5)\]

Simplify this step by step:

\[ y - 1 = -\frac{1}{3}x + \frac{5}{3} \]

Adding 1 to each side to isolate 'y': \[ y - 1 + 1 = -\frac{1}{3}x + \frac{5}{3} + 1 \]
\[ y = -\frac{1}{3}x + \frac{5}{3} + \frac{3}{3} \]
\[ y = -\frac{1}{3}x + \frac{8}{3} \]

Therefore, the y-intercept 'b' is \(\frac{8}{3}\).
simplifying equations
Simplifying equations is the process of rearranging and combining like terms to make the equation easier to solve or understand.

In our point-slope form: \[ y - 1 = -\frac{1}{3}(x - 5) \]

Distribute \( -\frac{1}{3} \) to terms inside the parenthesis:

\[ y - 1 = -\frac{1}{3}x + \frac{5}{3} \]

Add 1 to both sides to isolate 'y':

\[ y - 1 + 1 = -\frac{1}{3}x + \frac{5}{3} + 1 \]

Combine \(\frac{5}{3}\) and 1:

\[ y = -\frac{1}{3}x + \frac{8}{3} \]

The simplified slope-intercept form is \(y = -\frac{1}{3}x + \frac{8}{3}\). This form is \(y = mx + b\), clearly showing the slope and y-intercept.

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Most popular questions from this chapter

Solve each system using the substitution method. $$ \begin{aligned} &3 x+5 y=3\\\ &x=8-4 y \end{aligned} $$

Find the zero of each function. $$ f(x)=2 x+9 $$

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Remove parentheses and simplify. $$ 2 p-(7-4 p) $$

Solve each system using the elimination method. $$ \begin{aligned} x+3 y &=2 \\ 2 x-y &=11 \end{aligned} $$
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