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Chapter 1: Problem 32

Graph each line by hand. Give the \(x\) - and y-intercepts. $$4 x-3 y=9$$

Short Answer

Expert verified
The x-intercept is \(\left(\frac{9}{4}, 0\right)\) and the y-intercept is \((0, -3)\).

Step by step solution

01

Identify the Problem

We are given the linear equation \(4x - 3y = 9\). To graph this line, we need to find both the \(x\)-intercept and the \(y\)-intercept.
02

Find the x-intercept

To find the \(x\)-intercept, set \(y = 0\) in the equation and solve for \(x\):\[4x - 3(0) = 9\]\[4x = 9\]\[x = \frac{9}{4}\]So, the \(x\)-intercept is at \(\left(\frac{9}{4}, 0\right)\).
03

Find the y-intercept

To find the \(y\)-intercept, set \(x = 0\) in the equation and solve for \(y\):\[4(0) - 3y = 9\]\[-3y = 9\]\[y = -3\]Thus, the \(y\)-intercept is at \((0, -3)\).
04

Plot the Intercepts

Use the intercepts \(\left(\frac{9}{4}, 0\right)\) and \((0, -3)\) to plot points on the graph. These two points determine the line.
05

Draw the Line

Draw a straight line through the two intercept points to complete the graph of the equation.

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Key Concepts

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

Understanding x-intercepts
The x-intercept is a critical concept when graphing linear equations. It represents the point where a graph crosses the x-axis. At this point, the value of y is always zero.
  • To find the x-intercept, substitute 0 for y in the equation and solve for x. For example, in the equation \(4x - 3y = 9\), if we set \(y = 0\), we get:\[4x = 9\]
  • Simplifying yields \(x = \frac{9}{4}\), giving the intercept point as \(\left(\frac{9}{4}, 0\right)\).
  • This intercept is often plotted on the graph as it provides an anchor point.
Recognizing and calculating this point is essential for accurately graphing the line.
Understanding y-intercepts
Similar to the x-intercept, the y-intercept is where the graph crosses the y-axis, meaning the x value is zero at this point.
  • To find the y-intercept, substitute 0 for x in the equation and solve for y. For example, in the equation \(4x - 3y = 9\), substituting \(x = 0\) results in \(-3y = 9\).
  • Solving this equation gives \(y = -3\), indicating the intercept at \((0, -3)\).
  • This y-intercept is key for graphing because it serves as another crucial reference point.
Utilizing both intercepts allows for the drawing of a straight line, representing the linear equation on the graph.
The character of linear equations
A linear equation is any equation that can be written in the form of \(ax + by = c\), where a, b, and c are constants.
  • These equations consistently graph as a straight line when plotted on a coordinate plane.
  • The general characteristics of linear equations include a constant rate of change and a graph that doesn't curve or break.
  • Every linear equation has at least one solution and can have two intercepts, specifically the x- and y-intercepts.
Understanding these characteristics helps in recognizing the solutions' extent and how to approach plotting them.
Mastering graphing techniques for linear equations
Graphing linear equations efficiently is a fundamental skill in mathematics. It involves utilizing a few systematic techniques:
  • First, calculate both the x- and y-intercepts 鈥� these are the primary steps as they provide key points through which the line will pass.
  • Plotting these intercepts on the coordinate plane is crucial because they ease the visualization process.
  • Once plotted, simply draw a straight line through these points. Ensure the line extends in both directions to demonstrate that it continues infinitely.
  • Also, understanding the slope can help. It represents the steepness of the line, calculated as the change in y over the change in x.
By mastering these techniques, you'll efficiently map out linear equations, making them easier and faster to analyze.

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Most popular questions from this chapter

Solve each equation and inequality analytically. Use interval notation to write the solution set for each inequality. (a) \(5-3 x=x+1\) (b) \(5-3 x \leq x+1\) (c) \(5-3 x \geq x+1\)

Find the zero of the finction \(f\). Do not use a calculator. $$f(x)=1.5 x+2(x-3)+5.5(x+9)$$

Solve each inequality analytically, writing the solution set in interval notation. Support your answer graphically. (Hint: Once part (a) is done, the answer to part (b) follows.) (a) \(10 x+5-7 x \geq 8(x+2)+4\) (b) \(10 x+5-7 x<8(x+2)+4\)

When a bolt of lightning strikes in the distance, there is often a delay between seeing the lightning and hearing the thunder. The function \(f(x)=\frac{x}{5}\) computes the approximate distance in miles between an observer and a bolt of lightning when the delay is \(x\) seconds. (a) Find \(f(15)\) and interpret the result. (b) Graph \(y=f(x) .\) Let the domain of \(f\) be \([0,20]\).

If \(c\) is a zero of the linear function \(f(x)=m x+b, m \neq 0\), then the point at which the graph intersects the \(x\) -axis has coordinates (______,_______)
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