Problem 65 - Algebra 1

Question

Use a table of values to graph the equation. $$y=-x+5$$

Step-by-Step Solution

Answer

The graph of the equation \(y = -x + 5\) will be a straight line decreasing from the y-intercept at (0,5) and passing through points like (1,4), (2,3), and so forth.

Step 1: Identify the format

The given equation is in slope-intercept form (y = mx + c), where m is the slope and c is the y-intercept.

Step 2: Identify the slope and y-intercept

In the given equation \(y = -x + 5\), the slope (m) is -1 and the y-intercept (c) is 5. The slope indicates that for each step increase in x, y will decrease by 1. The y-intercept indicates that when x is zero, y is 5.

Step 3: Create a table of values

Choose several x values and find the corresponding y values using the equation. For example, when x is 0, y is 5 as per the y-intercept. Similarly, increase x by 1, y will decrease by 1.

Step 4: Plot the points on the graph

Draw the x and y axes and plot each (x, y) pair from the table of values. For example, plot the points (0,5), (1,4), (2,3), and so forth.

Step 5: Draw the line

After the points are plotted, draw a line through the points to create a graph of the equation.

Key Concepts

Slope-Intercept Form

When it comes to graphing linear equations, the slope-intercept form is one of the most convenient and commonly used representations. This form is written as y = mx + b, where m represents the slope of the line, and b represents the y-intercept. The slope tells us how steep the line is — in other words, how much the y value changes for a unit change in the x value. On the other hand, the y-intercept is the point where the line crosses the y-axis, indicating the value of y when x is zero.

To graph a line using the slope-intercept form, we start by plotting the y-intercept on the graph. Then, we use the slope to determine the direction and steepness of the line. If the slope is positive, the line rises from left to right; if negative, it falls. For instance, in the equation \( y = -x + 5 \), the slope is \( -1 \) which means for every step right on the x-axis, we move one step down on the y-axis. The y-intercept is \( 5 \), so the line will cross the y-axis at the point \( (0,5) \).

Understanding and identifying these components in the slope-intercept equation significantly simplifies graphing the line efficiently.

Table of Values

A table of values is a powerful tool in understanding how a linear equation behaves and assists in graphing it on a coordinate plane. The table lists a set of x values and the corresponding y values obtained by substituting the x values into the equation. Typically, you would choose a range of x values — which can include negative numbers, zero, and positive numbers — to get a clear picture of how the line trends across the graph.

When creating a table of values for the equation \(y = -x + 5\), you could start with \(x = 0\) which would give a y value of \(5\). Moving on, an increment in \(x\) by 1 to \(x = 1\) would decrease y by 1, according to the slope \(m = -1\), giving us \(y = 4\). By continuing this process and plotting the resulting coordinates, a clear pattern emerges which can be joined to form the straight line represented by the equation on the graph.

The key in using a table of values effectively is choosing x values that reflect the full range and nature of the line, ensuring that the graph you draw represents an accurate visualization of the equation.

Y-Intercept

In the equation of a line, the y-intercept is a fundamental component that helps determine the line's position on the graph. It is defined as the point where the line crosses the y-axis, which can be found when x is set to zero. The y-intercept is represented by the variable b in the slope-intercept form y = mx + b. This value provides a starting point for graphing the line and is particularly crucial when the graph does not pass through the origin, or the point (0,0).

For example, in our equation \(y = -x + 5\), the y-intercept is \(5\). This simply means that the point \( (0,5) \) is where the line will intersect the y-axis. When graphing, we can immediately plot this point without any calculations, as it is given directly by the equation. After plotting the y-intercept, we then use the slope to determine how the line will run across the graph. The y-intercept is essential because no matter how the rest of the line moves — up or down, left or right — it must pass through the y-intercept, anchoring the graph to this fixed point.

Special attention to the y-intercept while plotting can make the difference in accurately representing the line and adheres to the principle that the y-intercept is a non-negotiable reference point for the graph of a linear equation.