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Chapter 5: Problem 21

Write an equation of the line satisfying the given conditions. Passing through \((0,5)\) and \((5,2)\)

Short Answer

Expert verified
The equation of the line is \(y = -\frac{3}{5}x + 5\).

Step by step solution

01

- Find the slope

Use the formula for the slope, \(m\), between two points \(x_1, y_1\) and \(x_2, y_2\): \[m = \frac{y_2 - y_1}{x_2 - x_1}\]Substitute the given points \(0,5\) and \(5,2\):\[m = \frac{2 - 5}{5 - 0} = \frac{-3}{5}\]
02

- Use the point-slope form of the equation

Start with the point-slope form of the equation of a line: \[y - y_1 = m(x - x_1)\]Using \(m = -\frac{3}{5}\) and the point \(0,5\): \[y - 5 = -\frac{3}{5}(x - 0)\]
03

- Simplify the equation

Simplify the equation to the slope-intercept form, \(y = mx + b\): \[y - 5 = -\frac{3}{5}x\] Add 5 to both sides to isolate \(y\): \[y = -\frac{3}{5}x + 5\]

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Key Concepts

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

finding the slope
Understanding how to find the slope of a line is crucial in algebra. The slope represents how steep a line is. To calculate it, we use two points on the line.
The formula for the slope, which is usually denoted by the letter \(m\), is:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\text{, where }(x_1, y_1)\text{ and }(x_2, y_2)\text{ are the coordinates of the two points.}\]
For example, given points \((0,5)\) and \((5,2)\), we substitute the coordinates into the formula:
\[m = \frac{2 - 5}{5 - 0} = \frac{-3}{5}\]
So, the slope \(m\) is \(-\frac{3}{5}\). This means that for every 5 units you move to the right on the x-axis, you move 3 units down on the y-axis.
point-slope form
The point-slope form of a linear equation is a very convenient way to represent a line if you know a point on the line and its slope. The point-slope form is expressed as:
\[y - y_1 = m(x - x_1)\]
Here, \((x_1, y_1)\) is a known point on the line, and \(m\) is the slope.
For the line passing through \((0, 5)\) with a slope of \(-\frac{3}{5}\), we substitute into the point-slope form:
\[y - 5 = -\frac{3}{5}(x - 0)\]
This equation can be used directly to describe the line, or we can transform it to another form depending on our needs.
slope-intercept form
The slope-intercept form is one of the most common ways to write the equation of a line. It makes graphing straightforward because it explicitly shows the slope and the y-intercept of the line. The form is:
\[y = mx + b\]
Here, \(m\) is the slope, and \(b\) is the y-intercept, which is where the line crosses the y-axis.
To convert from point-slope form to slope-intercept form for our given problem, start with:
\[y - 5 = -\frac{3}{5}(x - 0)\]
Simplify and solve for \(y\):
\[y = -\frac{3}{5}x + 5\]
Now we have the line's equation in slope-intercept form, \(y = -\frac{3}{5}x + 5\), meaning the slope is \(-\frac{3}{5}\) and the y-intercept is 5.

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Most popular questions from this chapter

Solve each of the following problems algebraically. Be sure to label what the variable represents. A beaker contains \(40 \mathrm{ml}\) of a \(60 \%\) alcohol solution. What percentage of alcohol solution must be added to produce \(70 \mathrm{ml}\) of a \(45 \%\) solution?

Sketch the graph of the line satisfying the given conditions. Passing through \((-1,0)\) with slope 4

Sketch the graph of the line satisfying the given conditions. Passing through \((1,-2)\) with slope \(\frac{-3}{2}\)

A doctor uses a treadmill to administer cardiac stress tests to his patients. The treadmill is 5.5 feet long with a front end that can be raised to a maximum height of 10 inches. Find the maximum grade of the treadmill.

Determine the slope of the line from its equation. $$y=5 x+7$$
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