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Chapter 2: Problem 16

\(11-36\) Make a table of values and sketch the graph of the equation. Find the \(x\) - and \(y\) -intercepts. $$ x+y=3 $$

Short Answer

Expert verified
The x-intercept is at (3,0) and the y-intercept is at (0,3).

Step by step solution

01

Write the Equation in Slope-Intercept Form

The equation given is \(x + y = 3\). To write it in slope-intercept form, \(y = mx + b\), subtract \(x\) from both sides to isolate \(y\). This gives us \(y = -x + 3\). Now, it's in slope-intercept form with a slope of \(-1\) and a y-intercept of \(3\).
02

Create a Table of Values

Choose different values of \(x\) to find corresponding \(y\) values. Substitute \(x = 0, 1, 2\) into the equation \(y = -x + 3\).- For \(x = 0\), \(y = -0 + 3 = 3\).- For \(x = 1\), \(y = -1 + 3 = 2\).- For \(x = 2\), \(y = -2 + 3 = 1\).This gives us the table: \((0,3), (1,2), (2,1)\).
03

Sketch the Graph

Plot the points \((0,3), (1,2),\) and \((2,1)\) from the table onto a coordinate plane. Connect these points with a straight line, as the equation represents a linear function.
04

Determine the x-intercept

The x-intercept occurs where \(y = 0\). Set \(y = 0\) in the equation \(x + y = 3\), giving \(x + 0 = 3\). Thus, \(x = 3\) is the x-intercept, making the point \((3, 0)\).
05

Confirm and Note the y-intercept

The y-intercept was found in Step 1 as \((0, 3)\) by setting \(x = 0\) in the equation. This is confirmed by our table of values as well, and appears on the graph at the point where the line crosses the y-axis.

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form is a way to express the equation of a straight line. It makes it easy to identify the slope and y-intercept of the line, which are essential for graphing. The general form of this equation is \(y = mx + b\), where:
  • \(m\) represents the slope of the line
  • \(b\) represents the y-intercept, or the point where the line crosses the y-axis
To convert an equation like \(x + y = 3\) into slope-intercept form, rearrange it to solve for \(y\). By subtracting \(x\) from both sides, you can rewrite it as \(y = -x + 3\). This reveals that the slope \(m\) is \(-1\) and the y-intercept \(b\) is \(3\). The slope tells us how steep the line is, and a negative slope indicates that the line moves downwards from left to right.
Finding Intercepts
Intercepts are the points where a graph crosses the axes, and they are crucial for graphing equations. To find the x-intercept, set \(y = 0\) in your equation and solve for \(x\). In our equation \(x + y = 3\), this gives \(x + 0 = 3\). So, the x-intercept is at \(x = 3\), or the point \((3, 0)\).

Finding the y-intercept involves setting \(x = 0\) and solving for \(y\). With \(x + y = 3\), this results in \(0 + y = 3\), confirming the y-intercept \((0, 3)\). These intercepts help to plot the line accurately, as they show where the line intersects the axes.
Table of Values
Creating a table of values is a practical step in graphing a linear equation. It involves selecting various values for \(x\) and using the equation to find corresponding values of \(y\).

For example, using the slope-intercept form \(y = -x + 3\), we can substitute different values for \(x\) to build our table:
  • If \(x = 0\), then \(y = 3\)
  • If \(x = 1\), then \(y = 2\)
  • If \(x = 2\), then \(y = 1\)
With the ordered pairs \((0, 3), (1, 2), (2, 1)\), you have points that can be plotted on the graph. This provides a set of reference points that outline the line, making it easier to draw accurately.
Linear Graphing
Linear graphing involves representing equations as straight lines on a coordinate plane. It's a visual way of showing relationships between variables. With linear equations, each point on the line is a solution to the equation.

To graph the line for \(y = -x + 3\), you can plot the points obtained from your table of values. In this case, plot \((0, 3), (1, 2), (2, 1)\) on a coordinate plane. These points lie on a straight line. Drawing a line through them represents the equation graphically. Ensure the line is straight, indicating a consistent slope. Linear graphs are distinct in that they have no curves or bends, providing a clear visualization of the linear relationship.

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Most popular questions from this chapter

\(59-66\) . Find the solutions of the inequality by drawing appropriate graphs. State each answer rounded to two decimals. $$ x^{2} \leq 3 x+10 $$

Find the area of the triangle formed by the coordinate axes and the line $$ 2 y+3 x-6=0 $$

Collinear Points Suppose you are given the coordinates of three points in the plane and you want to see whether they lie on the same line. How can you do this using slopes? Using the Distance Formula? Can you think of another method?

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