Problem 93 - Algebra 1: Concepts and Skills

Question

Use a table to graph the equation. $$ x+y=-4 $$

Step-by-Step Solution

Answer

The line plotted by the points \((-2, -2)\), \((0, -4)\), \((2, -6)\) in the graph represents the equation \(x + y = -4\).

Step 1: Set up a table

A table for graphing an equation typically includes two columns: one for \(x\)-values and one for \(y\)-values. The general rule is to choose a few values for \(x\) (for instance, -2, 0, and 2), and then compute the corresponding \(y\) values using the equation.

Step 2: Calculate \(y\)-values

For each chosen \(x\)-value, substitute it into the equation and solve for \(y\). For instance, if \(x = -2\), then the equation becomes \(-2 + y = -4\), which simplifies to \(y = -4 - (-2) = -2\). Do this for each chosen \(x\)-value.

Step 3: Plot the points and draw the line

Using the values from the table, plot the points on the graph. Since this is a linear equation, the points should form a straight line. Connect the points with a straight line. The line represents all the solutions to the equation.

Key Concepts

Coordinate Graphing

Graphing is a visual way of representing mathematical concepts, and when it comes to linear equations, the coordinate plane is a vital tool. The coordinate plane consists of two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). By plotting points at the intersection of x and y values, we can visualize the relationship between them.

When graphing the linear equation \(x + y = -4\), you'll use this plane to place your points. Each point on the graph corresponds to a possible solution to the equation, with the x-coordinate representing the x value and the y-coordinate representing the y value. Once multiple points are plotted and connected, a picture of the equation's solutions emerges in the form of a straight line.

Linear Equation Solutions

A linear equation in two variables usually looks like \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. The solutions to a linear equation are all the x and y values that make the equation true. In our case, the equation \(x + y = -4\) has an infinite number of solutions because it represents a line on the coordinate plane.

The power of graphing comes from its ability to show every possible solution as a point on that line. No matter which point you choose from the line, the x and y coordinates will satisfy the equation. For instance, the point \(1, -5\) is on the line and is a solution because \(1 + (-5) = -4\).

Constructing a Table of Values

Before graphing, it's helpful to create a table of values. The table will guide you to plot the points accurately on the graph. Start by choosing a few x-values. Pick numbers that are easy to work with, like -2, 0, and 2. For each x-value, calculate the corresponding y-value using the equation.

For the equation \(x + y = -4\), let's construct a table:

• For \(x=-2\), we solve for y: \(-2 + y = -4\) yields \(y=-2\).
• For \(x=0\), y must be \(y=-4\).
• For \(x=2\), we get \(y=-6\).

These points, when plotted on the coordinate graph, will fall in a straight line, showing that they align with the linear equation given.