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Chapter 2: Problem 131

For the following exercises, sketch the graph of each equation. $$\frac{x}{3}-\frac{y}{4}=1$$

Short Answer

Expert verified
Plot intercepts (3,0) and (0,-4), then draw the line through them.

Step by step solution

01

Identify the Type of Equation

The equation \( \frac{x}{3} - \frac{y}{4} = 1 \) is a linear equation because both \(x\) and \(y\) appear to the first power.
02

Convert to Standard Form

To better understand the equation, rewrite it in the standard form of a line, \(Ax + By = C\). Multiply every term by 12 (the least common multiple of 3 and 4) to get \(4x - 3y = 12\).
03

Find the X-Intercept

To find the x-intercept, set \(y = 0\) in the equation \(4x - 3y = 12\). Solve for \(x\):\[4x - 3(0) = 12 \Rightarrow 4x = 12 \Rightarrow x = 3\]. So, the x-intercept is at \((3, 0)\).
04

Find the Y-Intercept

To find the y-intercept, set \(x = 0\) in the equation \(4x - 3y = 12\). Solve for \(y\): \[4(0) - 3y = 12 \Rightarrow -3y = 12 \Rightarrow y = -4\]. So, the y-intercept is at \((0, -4)\).
05

Plot the Intercepts

Plot the points \((3,0)\) and \((0,-4)\) on a coordinate plane. These are the points where the line crosses the x-axis and y-axis, respectively.
06

Draw the Line

Draw a straight line through the points \((3,0)\) and \((0,-4)\). Extend the line across the graph. This line represents the graph of the equation \( \frac{x}{3} - \frac{y}{4} = 1 \).

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Key Concepts

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

x-intercept
The x-intercept is the point where a line crosses the x-axis. This point can be found by setting the value of y in the equation to zero and solving for x. For this particular equation, when y is set to 0, we solve the equation to find x as follows:
\[4x - 3(0) = 12 \Rightarrow 4x = 12 \Rightarrow x = 3\]
So, the x-intercept is at the point (3, 0).
  • The x-axis is horizontal.
  • An x-intercept is found where the line meets the x-axis, meaning y is always 0 at this point.
In graphs, x-intercepts are important because they provide a precise horizontal location of the line in a coordinate plane. It's like a starting reference point when sketching graphs.
y-intercept
The y-intercept is where a line crosses the y-axis. You find this point by setting x equal to zero in the equation and solving for y. Let's see how it works with our equation:
\[4(0) - 3y = 12 \Rightarrow -3y = 12 \Rightarrow y = -4\]
Thus, the y-intercept occurs at (0, -4).
  • The y-axis stands vertically on a graph.
  • A y-intercept is the point where x is 0, highlighting the vertical intersection of the line with the y-axis.
Understanding the y-intercept is crucial as it provides contextual information on how a line behaves as it approaches the y-coordinate in a graph. This point helps in graphing the function and understanding its initial value in real-life applications.
standard form of a line
A linear equation can often be rearranged into the standard form, which is \(Ax + By = C\). This structure helps quickly identify equations of lines, making it a valuable format in algebra. In our example, we began with \(\frac{x}{3} - \frac{y}{4} = 1\) and transformed it to:
\[4x - 3y = 12\]
  • A, B, and C should be integers.
  • This form allows for easy identification and solving of both intercepts.
The standard form is particularly effective when working with systems of equations or when trying to quickly graph equations on the Cartesian plane, as it displays the fundamental structure of linear relationships clearly.

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Most popular questions from this chapter

For the following exercises, use each set of data to calculate the regression line using a calculator or other technology tool, and determine the correlation coefficient to 3 decimal places of accuracy. $$\begin{array}{|c|c|c|c|c|c|c|}\hline x & {900} & {988} & {1000} & {1010} & {1200} & {1205} \\ \hline y & {70} & {80} & {82} & {84} & {105} & {108} \\\ \hline\end{array}$$

Graph \(f(x)=0.5 x+10 .\) Pick a set of 5 ordered pairs using inputs \(x=-2,1,5,6,9\) and use linear regression to verify that the function is a good fit for the data.

Determine whether the following function is increasing or decreasing. \(f(x)=7 x+9\)

For the following exercises, use each set of data to calculate the regression line using a calculator or other technology tool, and determine the correlation coefficient to 3 decimal places of accuracy. $$\begin{array}{|c|c|c|c|c|c|c|c|c|}\hline x & {3} & {4} & {5} & {6} & {7} & {8} & {9} & {10} \\ \hline y & {21.9} & {22.22} & {22.74} & {22.26} & {20.78} & {17.6} & {16.52} & {18.54} \\ \hline x & {11} & {12} & {13} & {14} & {15} & {16} & {17} & {18} \\ \hline y & {15.76} & {13.68} & {14.1} & {14.02} & {11.94} & {11.94} & {11.28} & {9.1} \\ \hline\end{array}$$

For the following exercises, consider this scenario: The profit of a company decreased steadily over a ten-year span. The following ordered pairs shows dollars and the number of units sold in hundreds and the profit in thousands of over the tenyear span, (number of units sold, profit) for specific recorded years: $$(46,1,600), \quad(48,1,550), \quad(50,1,505), \quad(52,1,540), \quad(54,1,495).$$ Use linear regression to determine a function \(P\) where the profit in thousands of dollars depends on the number of units sold in hundreds.
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