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Chapter 4: Problem 80

In the following exercises, graph by plotting points. \(x+y=4\)

Short Answer

Expert verified
Plot points (4, 0), (0, 4), (2, 2). Then draw a line through them.

Step by step solution

01

Understand the Equation

The equation given is a linear equation in two variables: \[x + y = 4\]
02

Find Intercepts

To graph the equation, first find the x-intercept and the y-intercept.For the x-intercept, set \(y = 0\): \[x + 0 = 4 \ x = 4\]So, the x-intercept is \((4, 0)\).For the y-intercept, set \(x = 0\): \[0 + y = 4 \ y = 4\]So, the y-intercept is \((0, 4)\).
03

Choose Additional Points

Choose additional values for x and solve for y to get more points. For example, set \(x = 2\): \[2 + y = 4 \ y = 2\]So, another point is \((2, 2)\).
04

Plot the Points

Plot the points found on a graph: \((4, 0), (0, 4), (2, 2)\).
05

Draw the Line

Draw a straight line through the points to represent the equation \(x + y = 4\). This line is 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.

Plotting Points
Plotting points is an essential skill in graphing linear equations. To plot a point, you need an ordered pair \((x, y)\) that locates the point on the grid. The first number in the pair is the x-coordinate, which tells you how far to go left or right from the origin. The second number is the y-coordinate, which informs you how far up or down to move. For example, the point \(4, 0\) means moving 4 units to the right and 0 units up or down. Always start from the origin \(0, 0\) when plotting points.
Linear Equation
A linear equation like \(x + y = 4\) forms a straight line when graphed. This type of equation has variables raised only to the power of one. The simplest form of a linear equation is \(y = mx + b\), where \m\ is the slope and \b\ is the y-intercept. Every solution of a linear equation produces a point on the line, and this is why plotting points helps visualize the relationship between the two variables.
Intercepts
Intercepts are the points where a line crosses the x-axis or y-axis. To find the x-intercept, set \ y = 0 \ and solve for \ x \. For the y-intercept, set \ x = 0 \ and solve for \ y \. In the example \( x + y = 4 \), the x-intercept is found by setting \ y = 0 \ which gives \ x = 4 \. Thus, the x-intercept is \(4, 0\). Similarly, the y-intercept is found by setting \ x = 0 \, resulting in \ y = 4 \ or \(0, 4\). These intercepts are crucial points for drawing the line of the equation.
Graphing
Graphing involves plotting points and drawing a line through them to represent an equation visually. In our example, start by plotting the intercepts \(4, 0\) and \(0, 4\). Add intermediate points like \(2, 2\) for better accuracy. Place these points on the graph and draw a straight line through them. This line represents all solutions to the equation \( x + y = 4 \). By connecting the dots, you illustrate the linear relationship and make it easier to understand the equation's behavior.

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