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Magazine Chapter 11: Problem 336
In the following exercises, graph the line given a point and the slope. $$ (2,-2) ; m=\frac{5}{2} $$
Short Answer
Expert verified
Graph the line passing through \(2, -2\) with slope \(\frac{5}{2}\). Equation: \(y = \frac{5}{2}x - 7\).
Step by step solution
01
Understand the Problem
Given a point \(2, -2\) and a slope \(m = \frac{5}{2}\), you need to graph the equation of the line that passes through the given point with the given slope.
02
Recall the Slope-Intercept Form
The slope-intercept form of a line is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
03
Use the Point-Slope Formula
Since a point and the slope are given, use the point-slope formula \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \( (x_1, y_1) \) is the given point. Substituting the values, it becomes \(y + 2 = \frac{5}{2}(x - 2)\).
04
Simplify to Slope-Intercept Form
Simplify the equation to get it into the slope-intercept form \(y = mx + b\). Starting with \(y + 2 = \frac{5}{2}(x - 2)\), distribute \(\frac{5}{2}\): \[y + 2 = \frac{5}{2}x - 5\]. Then, isolate \(y\) by subtracting 2: \[y = \frac{5}{2}x - 5 - 2\]. Hence, \[y = \frac{5}{2}x - 7\].
05
Plot the Point
Start by plotting the given point \(2, -2\) on the coordinate plane.
06
Use the Slope to Find Another Point
From the point \(2, -2\), use the slope \( \frac{5}{2} \). This means from \(x = 2\), move 2 units to the right (making \(x = 4\)) and move 5 units up (making \(y = 3\)). So, another point on the line is \(4, 3\).
07
Draw the Line
Draw a line through the points \(2, -2\) and \(4, 3\). This is the graph of the line with the given point and slope.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Point-Slope Form
To graph a line when you have a point and the slope, you'll often use the point-slope form of a linear equation. This form is expressed as: \[y - y_1 = m(x - x_1)\] Here,
- \((x_1, y_1)\) is a point on the line
- \(m\) is the slope
Slope-Intercept Form
The slope-intercept form is another way to express the linear equation, which can be quite useful when graphing. It鈥檚 written as: \[y = mx + b\] Here's what each term stands for:
- \(m\) is the slope of the line
- \(b\) is the y-intercept, or where the line crosses the y-axis
Coordinate Plane
Understanding the coordinate plane is crucial for graphing linear equations. The coordinate plane consists of two perpendicular lines that intersect at the origin \((0,0)\):
- The horizontal axis is called the x-axis
- The vertical axis is called the y-axis
- A point \((2, -2)\) means moving 2 units to the right from the origin and 2 units down.
Plotting Points
Plotting points is the foundational step for graphing lines. Here鈥檚 a step-by-step guide:
1. **Identify the Point:** Start with the given point, in our case, \((2, -2)\).
2. **Locate on Coordinate Plane:** Find \(2\) on the x-axis and \(-2\) on the y-axis. This pinpoints your first point.
3. **Plot Additional Points Using Slope:** Use the slope to find more points. For example, with slope \( \frac{5}{2} \):
1. **Identify the Point:** Start with the given point, in our case, \((2, -2)\).
2. **Locate on Coordinate Plane:** Find \(2\) on the x-axis and \(-2\) on the y-axis. This pinpoints your first point.
3. **Plot Additional Points Using Slope:** Use the slope to find more points. For example, with slope \( \frac{5}{2} \):
- From \(x = 2\), move 2 units right (making \(x = 4\)) and
- Move 5 units up (making \(y = 3\))
