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Chapter 3: Problem 75

(a) find three solutions of the equation. (b) graph the equation. \(y=-\frac{1}{3} x+5\)

Short Answer

Expert verified
Solutions: (0, 5), (3, 4), (-3, 6). Graph: a straight line through these points.

Step by step solution

01

Understanding the Equation

The equation given is in slope-intercept form: \(y = mx + b\). In this case, \(m = -\frac{1}{3}\) and \(b = 5\).
02

Find the Y-Intercept

Since \(b = 5\), the y-intercept is (0, 5). This is one of the three solutions.
03

Find Additional Points

Choose another value for \(x\). If \(x = 3\):
04

Calculate Y for \(x = 3\)

Substitute \(x = 3\) into the equation: \(y = -\frac{1}{3}(3) + 5 = -1 + 5 = 4\). So, another solution is (3, 4).
05

Find One More Point

Choose another value for \(x\). If \(x = -3\):
06

Calculate Y for \(x = -3\)

Substitute \(x = -3\) into the equation: \(y = -\frac{1}{3}(-3) + 5 = 1 + 5 = 6\). Another solution is (-3, 6).
07

Graph the Equation

Plot the three points (0, 5), (3, 4), and (-3, 6) on a graph. Draw a straight line through these points, which represents the equation \(y = -\frac{1}{3}x + 5\).

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Key Concepts

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

slope-intercept form
The slope-intercept form is a way of writing linear equations. The general format is: \( y = mx + b \).
  • m represents the slope of the line.
  • b represents the y-intercept, where the line crosses the y-axis.
In our exercise, the equation is given as \( y = -\frac{1}{3}x + 5 \). Here, the slope \( m \) is \( -\frac{1}{3} \) and the y-intercept \( b \) is 5.

Understanding the slope-intercept form helps in quickly identifying these critical components. This makes graphing and solving equations easier. The y-intercept \( b \) gives the starting point on the y-axis, and the slope \( m \) tells how steep the line is. If \( m \) is negative, the line slopes downwards.
finding solutions
Finding solutions for a linear equation involves choosing values for \( x \) and calculating the corresponding \( y \) values. This helps plot points that lie on the line represented by the equation.

For the given equation \( y = -\frac{1}{3}x + 5 \), let's find three solutions:
  • First solution: The y-intercept is a natural choice. When \( x = 0 \), \( y = 5 \), so one solution is (0, 5).
  • Second solution: Choose \( x = 3 \). Substitute it into the equation: \( y = -\frac{1}{3}(3) + 5 = -1 + 5 = 4 \). Hence, another solution is (3, 4).
  • Third solution: Choose \( x = -3 \). Substitute it into the equation: \( y = -\frac{1}{3}(-3) + 5 = 1 + 5 = 6 \). Hence, one more solution is (-3, 6).
By finding these points, we set up a guide for the next step, which is graphing the equation.
graphing equations
Graphing linear equations allows you to visually see the relationship between \( x \) and \( y \) values.

To graph the equation \( y = -\frac{1}{3}x + 5 \), follow these steps:
  • Step 1: Find the y-intercept, which is (0, 5). Plot this point on the graph.
  • Step 2: Choose another \( x \) value (e.g., \( x = 3 \)) and find \( y \). Here, \( y = 4 \), so plot this point (3, 4).
  • Step 3: Choose one more \( x \) value (e.g., \( x = -3 \)) and find \( y \). Here, \( y = 6 \), so plot this point (-3, 6).
  • Step 4: Draw a straight line through the plotted points. This line represents the equation \( y = -\frac{1}{3} x + 5 \).


The graph is a straight line because it's a linear equation. Each point plotted is a solution, showing how \( y \) changes with \( x \). By understanding the slope and intercept, you can quickly sketch the graph and analyze the equation visually.

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Most popular questions from this chapter

(a) write the equation of the vertical line that passes through the point. (b) graph the equation. \((-2,-1)\)

For exercises 37-66, use the slope formula to find the slope of the line that passes through the points. \((1,9) ;(3,14)\)

(a) write the equation of the vertical line that passes through the point. (b) graph the equation. \((6,3)\)

(a) find three solutions of the equation. (b) graph the equation. \(y=10 x+50\)

(a) graph the given points, and draw a line through the points. (b) use the graph to find the slope of the line. (c) use the slope formula to find the slope of the line. \((-20,-50) ;(10,30)\)
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