Problem 66 - Algebra 1

Question

Use a table of values to graph the equation. $$y=x-7$$

Step-by-Step Solution

Answer

The graph of the equation \(y=x-7\) is a straight line that passes through the points (-2,-9), (0,-7), (2,-5), (4,-3), and (6,-1). It has a slope of 1 and crosses the y-axis at y=-7.

Step 1: Setting up the table of values

Choose five values for x. The values can be arbitrary, but it's often useful to include negative, zero, and positive values. For this exercise let's use -2, 0, 2, 4, and 6. Then, apply these values into the equation to find the corresponding y-values.

Step 2: Calculating y-values

Now, substitute each chosen x-value into the equation and solve for y. For example, when x=-2, \(y=-2-7=-9\). Repeat for each x-values. The result should be this table: \n\n|x | y |\n|--- |--- |\n|-2 |-9|\n|0|-7 |\n|2|-5 |\n|4|-3|\n|6|-1|

Step 3: Plotting the points

Each (x, y) pair represents a point in a two-dimensional space. So, for each pair, put a point in the corresponding position on the coordinate grid. In the end, there should be five points.

Step 4: Drawing the graph

Finally, connect all the points with a straight line to complete the graph of the equation \(y=x-7\). The line should cross through all five points, showing a linear relationship between x and y.

Key Concepts

Table of Values

A table of values is a helpful tool when graphing linear equations. It allows you to organize and calculate the outputs of an equation based on different input values. To create a table of values, you start by selecting a handful of values for the independent variable, usually represented as \(x\).

In our example with the equation \(y = x - 7\), we chose values \(-2, 0, 2, 4,\) and \(6\). These values are substituted into the equation to find the dependent variable \(y\).

• For \(x = -2\), \(y = -9\).
• For \(x = 0\), \(y = -7\).
• For \(x = 2\), \(y = -5\).
• For \(x = 4\), \(y = -3\).
• For \(x = 6\), \(y = -1\).

With this table of values, you have a set of coordinates \((x, y)\) ready to be plotted on a graph. By computing these pairs, a clear pattern might emerge, showing how each input affects the outcome.

Coordinate Grid

A coordinate grid is like a map for your graph. It comprises two axes, the horizontal \(x\)-axis and the vertical \(y\)-axis. Each point on the grid is defined by a pair of numbers known as coordinates. These coordinates are in the form \((x, y)\), dictating where a point should be placed on the grid.

To plot a point, you start at the origin, which is \((0,0)\), the center where the axes intersect. You then move horizontally to the \(x\)-coordinate and vertically to the \(y\)-coordinate.

Using our earlier example, plot each point from the table of values:

• Start at the origin.
• Move \(x\) units along the \(x\)-axis. For example, for \((-2, -9)\), move left to \(-2\).
• Then, move \(y\) units up or down for the \(y\)-coordinate, like moving down to \(-9\).

After plotting all the points on the grid, you'll have a visual representation that can help you see the relationship between \(x\) and \(y\).

Linear Relationship

A linear relationship shows that two variables are proportionally linked, forming a straight line when graphed on a coordinate grid. The equation \(y = x - 7\) is a classic example of a linear equation.

In mathematics, any equation that can be represented by the formula \(y = mx + c\) is considered linear. Here, \(m\) is the slope of the line, indicating the steepness and direction of the line. The \(c\) represents the \(y\)-intercept, the point where the line crosses the \(y\)-axis.

For the equation \(y = x - 7\), the slope \(m\) is 1, meaning the line rises at a consistent rate as \(x\) increases. The \(y\)-intercept is \(-7\), indicating it crosses the \(y\)-axis at \( (0, -7) \).

Understanding the linear relationship helps predict the change in \(y\) as \(x\) varies, making it easier to sketch the graph even without a table of values. Knowing this makes graphing linear equations quicker and more intuitive.