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Chapter 5: Problem 27

Sketch the graph of the given equation. Label the intercepts. $$3 x-4 y=12$$

Short Answer

Expert verified
Convert to \(y = \frac{3}{4}x -3\). Plot (0, -3) and (4, 0). Draw and label the line.

Step by step solution

01

- Convert the Equation to Slope-Intercept Form

Start by converting the given linear equation into the slope-intercept form, which is useful for graphing. Use the equation provided: \(3x - 4y = 12\)Rearrange to isolate y:\(3x - 12 = 4y$$\frac{3x - 12}{4} = y\)This simplifies to:\(y = \frac{3}{4}x - 3\)Now the equation is in the slope-intercept form: \(y = mx + b\), where \(m = \frac{3}{4}\) and \(b = -3\).
02

- Identify the Y-intercept

In the slope-intercept form \(y = mx + b\), \(b\) (the y-intercept) is the point where the line crosses the y-axis. For this equation \(b = -3\), so the y-intercept is at the point (0, -3).
03

- Identify the X-intercept

To find the x-intercept, set \(y\) to 0 in the original equation and solve for \(x\):\(3x - 4(0) = 12$$3x = 12$$x = 4\)So, the x-intercept is at the point (4, 0).
04

- Plot the Intercepts

On a graph, mark the points (0, -3) and (4, 0). These are the y-intercept and x-intercept respectively.
05

- Draw the Line

Connect these two points with a straight line. Since they are intercepts of a linear equation, the line extends infinitely in both directions through the points.
06

- Label the Intercepts

Label the points (0, -3) and (4, 0) on the graph to clearly indicate the intercepts.

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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 popular way to express linear equations, making it straightforward to graph them. It is written as:
\[ y = mx + b \]
Here,
  • **m** is the slope of the line, indicating how steep the line is. It shows how much y changes for a unit change in x.
  • **b** is the y-intercept, which is where the line crosses the y-axis. This tells us the value of y when x is 0.
To graph a line, having the equation in this form is incredibly useful. From our exercise example, the equation \[ 3x - 4y = 12 \] was rearranged to be in slope-intercept form. This conversion helps us understand the line's behavior better by directly giving the slope and y-intercept.
Plotting Intercepts
When graphing linear equations, **plotting intercepts** is a crucial step. Intercepts are where the line crosses the axes, which we can use to easily determine two points on the line. To clarify,
  • **Y-intercept**: This is where the line crosses the y-axis. It can be found by setting x to 0 in the equation and solving for y.
  • **X-intercept**: This is where the line crosses the x-axis. It can be found by setting y to 0 in the equation and solving for x.
For instance, in our example, we identified the y-intercept by noticing that \[ b = -3 \] in the slope-intercept form of the equation, meaning the y-intercept is at (0, -3).
Similarly, we identified the x-intercept by solving the original equation with y set to 0, giving us the point (4, 0). These points are instrumental in accurately graphing the linear equation.
Introduction to Linear Equations
**Linear equations** are equations that, when graphed, form a straight line. They are defined by the general form \[ Ax + By = C \] where
  • **A**,
  • **B**,
  • **C**
are constants. These equations can be easily transformed into slope-intercept form, making it easier to understand and graph them.
Linear equations have remarkable properties:
  • **They have constant slopes**: This means the rate of change between y and x is consistent throughout.
  • **They have unique intercepts**: Each linear equation will cross the x and y axes at specific points, unless the equation runs parallel to an axis.
  • **Easy to predict**: Because of their consistent rate of change, predicting values is straightforward once the equation is known.
Graphing a linear equation is part of understanding its properties. By converting it to the slope-intercept form, identifying intercepts, and plotting them, you'll have a clear visual representation of the linear relationship.

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Most popular questions from this chapter

A computer store budgets \(\$ 12,000\) to buy computers and laser printers. Each computer costs \(\$ 650\) and each printer costs \(\$ 200\). (a) Let \(C\) represent the number of computers and \(P\) represent the number of printers. 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 the shipment contains 16 computers, use the equation you obtained in part (a) to find the number of printers.

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

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 \(f=0.1\)

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. $$\begin{array}{|l|c|c|c|} \hline x \text { (Number of clerks working) } & 6 & 8 & 9 \\ \hline y \text { (Number of minutes waiting time) } & 8 & 5 & 3.5 \\ \hline \end{array}$$ What would the waiting time be if 4 clerks are working?

Solve each of the following problems algebraically. Be sure to label what the variable represents. On a certain history exam, \(46 \%\) of the questions were multiple-choice questions, 5 were essay questions, and the remaining \(44 \%\) were fill-in-the- blank questions. How many multiple-choice questions were there?
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