Problem 16 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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 \(3\).

Step 2: 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: \((0,3), (1,2), (2,1)\).

Step 3: 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.

Step 4: 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 \((3, 0)\).

Step 5: 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.

Key Concepts

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.