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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.

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 \(y\).

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.