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Chapter 12: Problem 33

Graph the equation of a straight line that passes through \((-1,3)\) and has a slope of -3 .

Short Answer

Expert verified
The equation of the line is y = -3x.

Step by step solution

01

- Substitute Point into Slope-Intercept Form

The slope-intercept form of a line is given by y = mx + b where m is the slope and b is the y-intercept. We are given a slope (m = -3) and a point (-1, 3). Use these to solve for b by substituting the values into the equation.3 = -3(-1) + b
02

- Solve for y-intercept

Simplify the equation to find b:3 = 3 + b Subtract 3 from both sides:0 = b So, the y-intercept (b) is 0.
03

- Write the Equation of the Line

With m = -3 and b = 0, use the slope-intercept form to write the equation of the line:y = -3x + 0 This simplifies to:y = -3x
04

- Graph the Line

To graph the equation y = -3x, start with the y-intercept (0,0). From this point, use the slope to plot the next point. Since the slope is -3, go down 3 units and right 1 unit to plot another point at (1, -3). Draw a straight line through these points.

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Key Concepts

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

Understanding Slope-Intercept Form
The slope-intercept form is a common way to express the equation of a straight line. The formula is: \( y = mx + b \).
In this form:
  • \(m\) is the slope
  • \(b\) is the y-intercept
The slope tells you how steep the line is and the direction it goes (up or down). The y-intercept tells you where the line crosses the y-axis.
For example, in the equation \( y = -3x + 0 \), the slope \(m\) is -3 and the y-intercept \(b\) is 0.
Finding the Y-Intercept
To find the y-intercept from an equation, you substitute the values you have into the slope-intercept form.
In this exercise, we started with the point \((-1, 3)\) and the slope \(-3\). Plugging these into the equation \( y = mx + b \), you get:
3 = -3(-1) + b
This simplifies to 3 = 3 + b. By subtracting 3 from both sides, you get \( b = 0 \).
So, the y-intercept is 0. This means the line crosses the y-axis at the origin, \((0, 0)\).
Graph Plotting
The final step is to graph the equation on a coordinate plane.
Start at the y-intercept, which is \((0, 0)\).
Using the slope, move down 3 units and to the right 1 unit to plot the next point.
So, from \((0, 0)\), you end up at \((1, -3)\).
Plot these points, then draw a straight line through them.
Remember, the slope represents the rise over run. A negative slope means the line goes down as you move from left to right. A positive slope means it goes up. By following these steps, you can plot any linear equation accurately.

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Most popular questions from this chapter

What is the equation for a line perpendicular to \(6 x-7 y=8\) and passing through the point \((8,1)\) ? A. \(y=-\frac{7}{6} x+\frac{31}{3}\) B. \(y=\frac{6}{7} x-\frac{8}{7}\) C. \(-6 x+7 y=-8\) D. \(y=\frac{7}{6} x-\frac{31}{3}\)

Graph the equation \(5 x-4 y=-20\).

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