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Chapter 8: Problem 113

Graph: \(4 x-5 y=20\)

Short Answer

Expert verified
The graphical representation of the equation produces a straight line passing through two points (0, -4) and (5, 4).

Step by step solution

01

Rearrange Equation

The given equation is \(4 x-5 y=20\). Transportation of the equation to get 'y' in terms of 'x' will yield \(y = (4x - 20)/5\).
02

Substitution

Now select two or more values of 'x', and for each selected 'x' substitute it in the equation to get corresponding 'y' value. For instance, choose 'x' as 0 and 5. Using the rearranged equation we can calculate the 'y' values: If 'x' = 0, then 'y' = -4, and if 'x' = 5, then 'y' = 4.
03

Plot the points

Next, plot these points (0, -4) and (5, 4) on the graph. After plotting the points, draw a straight line through these two points.
04

Extend the line

Extend the line on both ends to cover the entire graphical representation. This line represents the solution to the equation.

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Key Concepts

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

Graphing Linear Equations
Graphing a linear equation involves representing it as a straight line on a graph. To start, transform the equation into a more graph-friendly format, usually solving for one variable in terms of another (like y in terms of x). This allows us to identify points that lie on the line, which is essential for plotting. We can then choose a few values for x to find corresponding y-values using the rearranged equation.
  • Select any values for x.
  • Substitute these into the equation to solve for y.
  • Use the pairs (x, y) to plot on a graph.
Once enough points are plotted, connect them with a straight line. This line graphically represents all solutions to the equation. Adjust the line to ensure it extends across the graph, showing the complete linear relationship.
Understanding the Coordinate System
The coordinate system is a key tool to visualize mathematical equations. It consists of two axes: the x-axis (horizontal) and the y-axis (vertical). They intersect at a point called the origin, denoted by (0, 0).
The coordinates are noted as (x, y), where:
  • x is the horizontal position.
  • y is the vertical position.
Plotting points require these references to be located accurately on the graph. Each point on the graph is derived from these coordinates, making it easier to precisely plot and connect them for any linear equation.
Rearranging Equations
Rearranging equations is a pivotal step in graphing linear equations. The goal is to express one variable in terms of the other, such as rewriting an equation from a general form, like Ax + By = C, to a slope-intercept form, y = mx + b.
  • Focus on isolating one variable on one side of the equation.
  • This often involves basic algebraic operations: addition, subtraction, multiplication, or division.
By converting to y = mx + b, it becomes easy to identify the slope (m) and y-intercept (b). This standard form simplifies the graphing process by providing direct insights into how to plot the line, selecting points and identifying the linear path on the graph itself.

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Most popular questions from this chapter

Without using a calculator and knowing that \(\sqrt{2} \approx 1.4142\) rationalizing the denominator of \(\frac{1}{\sqrt{2}}\) makes division to obtain a decimal approximation for \(\frac{1}{\sqrt{2}}\) easier to perform. Because 10 and 8 share a common factor of \(2,\) I simplified \(\frac{\sqrt{10}}{8}\) to \(\frac{\sqrt{5}}{4}\)

$$\text { Simplify: } \sqrt{13+\sqrt{2}+\frac{7}{3+\sqrt{2}}}$$

In Exercises \(53-74\), rationalize each denominator. Simplify, if possible. $$\frac{8}{\sqrt{7}+\sqrt{3}}$$

In Exercises \(94-97,\) determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{3 \sqrt{x}}{x \sqrt{6}}=\frac{\sqrt{6 x}}{2 x} \text { for } x>0$$

In Exercises \(53-74\), rationalize each denominator. Simplify, if possible. $$\frac{2}{4+4-\sqrt{x}}$$
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