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Chapter 5: Problem 33

Sketch the graph of the line satisfying the given conditions. Passing through \((1,3)\) with slope 3

Short Answer

Expert verified
The line's equation is \( y = 3x \). Sketch a line through the origin with a slope of 3.

Step by step solution

01

- Identify the Slope-Intercept Form

The slope-intercept form of a linear equation is given by the formula: \[ y = mx + b \]where \( m \) is the slope and \( b \) is the y-intercept.
02

- Plug in the Slope

Given that the slope \( m = 3 \), the equation becomes: \[ y = 3x + b \]
03

- Use the Given Point

The line passes through the point \((1, 3)\). Substitute \(x = 1\) and \(y = 3\) into the equation to find \(b\): \[ 3 = 3(1) + b \]
04

- Solve for the Y-Intercept

Solve the equation \[ 3 = 3 + b \] for \(b\): \[ 3 - 3 = b \]\[ b = 0 \]
05

- Write the Final Equation

With \(b = 0\), the final equation of the line is: \[ y = 3x \]
06

- Sketch the Graph

To sketch the graph, start at the origin \((0, 0)\), which is the y-intercept, and use the slope \(m = 3\) to plot another point. From \((0,0)\), move 1 unit right (positive direction on the x-axis) and 3 units up (positive direction on the y-axis) to reach another point \((1, 3)\). Draw the line passing through these points.

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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 of a linear equation is a simple and powerful way to write the equation of a line. The formula is: The slope indicates how steep the line is, while the y-intercept tells you where the line crosses the y-axis. This form is great for quickly plotting lines and understanding their behavior.
Here's the formula again:
y = mx + b
where:
m = slope of the line
b = y-intercept
finding the y-intercept
Finding the y-intercept is simple when you have a point through which the line passes and the slope.
First, substitute the given point and slope into the slope-intercept form equation:
When the line passes through (x,y) = (1, 3) and has a slope of 3, the equation looks like this:
3 = 3*(1) + b
Now solve for b by isolating it:
3 = 3 + b
0 = b
So, the y-intercept b is 0. This means our line passes through the origin. This intercept shows where the line crosses the y-axis.
plotting points
Once you have the slope and y-intercept, plotting points becomes easy.
Start with the y-intercept. In our example, b = 0. So the first point is (0, 0).
Next, use the slope to find more points. The slope m = 3 means the line rises 3 units for every 1 unit it moves to the right.
From (0,0), move 1 unit to the right (positive x-direction) and 3 units up (positive y-direction). This gives you the point (1, 3).
Keep plotting points using these steps to draw the line. If needed, move in the opposite directions to plot more points, for instance moving 1 unit left and 3 units down to get another point like (-1, -3).
Connecting these points will give you the accurate graph of the line.
linear equations
Linear equations represent lines on a graph. These equations form straight lines and follow a predictable pattern.
They take the general form y = mx + b.
Here鈥檚 a breakdown:
- m is the slope, showing how steep the line is.
- b is the y-intercept, showing where the line crosses the y-axis.
Linear equations are easy to work with and provide valuable insight into relationships between variables.
They commonly appear in algebra and real-life applications like calculating speed or predicting trends.
Understanding linear equations is essential for mastering more complex mathematical concepts.

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Most popular questions from this chapter

Round off to the nearest hundredth when necessary. A sidewalk food vendor knows that selling 80 franks costs a total of 73 dollar and selling 100 franks costs a total of 80 dollar Assume that the total cost \(c\) of the franks is linearly related to the number \(f\) of franks sold. (a) Write an equation relating the total cost of the franks to the number of franks sold. (b) Find the cost of selling 90 franks. (c) How many franks can be sold for a total cost of 90.50 dollar (d) The costs that exist even if no items are sold are called the fixed costs. Find the vendor's fixed costs. I Hint: If no franks are sold then \(f=0.1\)

An electronics discount store wants to use up a credit of \(\$ 9,110\) with its supplier to order a shipment of VCRs and TVs. Each VCR costs \(\$ 125\) and each TV costs \(\$ 165\) (a) Let \(v\) represent the number of VCRs and \(t\) represent the number of TVs. Write an equation that reflects the given situation. (b) Sketch the graph of this relationship. Be sure to label the coordinate axes clearly. (c) If 28 VCRs are ordered, use the equation you obtained in part (a) to find the number of TVs.

Simplify the given expression. $$x^{2}\left(x^{3}\right)\left(x^{2}\right)^{3}$$

Sets of values are given for variables having a linear relationship. In each case, write the slope-intercept form for the equation of the line corresponding to the given set of values and answer the accompanying question. $$\begin{array}{|l|c|c|c|} \hline x \text { (Number of people waiting in line) } & 6 & 8 & 9 \\ \hline y \text { (Number of minutes waiting time) } & 10 & 15 & 17.5 \\ \hline \end{array}$$ What would the waiting time be if 15 people are waiting in line?

Determine the slope of the line from its equation. $$y=-4 x+2$$
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