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What To Remember When Graphing Linear Equations

graphing linear equations: Cartesian graph

Graphing linear equations is key to helping you with making observations or insights about the equation. Before we look at graphing linear equations, let's review graphing basics.

The image above is a Cartesian graph. The x-axis goes left to right, and the x-intercept is wherever a straight line crosses over or intersects this horizontal axis. The y-axis goes up and down, and the y-intercept is wherever a straight line crosses over or intersects this vertical axis. Think of this as a rectangular coordinate system formed with evenly spaced perpendicular lines. Each ordered pair of coordinates (x,y) denotes a singular point on the graph.

The graph of a linear equation can be drawn by finding which x values and y values solve for the equation of the line, meaning those values balance the equation. For example, (2, 5) is a valid graph point for the equation graphing linear equations: y=2x+1 formula, because substituting 2 for x and 5 for y in the equation gives 5 = 2(2)+1, or 5=5. (0, 0) is not a solution because 0≠0+1, or 0≠1, and therefore is not a point on the line. Once you have collected several points, you can graph your equation.

Find the x-coordinate of a point on the graph by counting the number of spaces along the x-axis in accordance with the first value. Then, find the y-coordinate by looking at the second number and counting that number of spaces up or down the y-axis. For example, the point (2,3) would be two spaces to the right and three spaces up.

Linear Equations

A linear equation, sometimes called a linear function, is an equation that forms a straight line when graphed. The graph of the equation can be displayed on horizontal and vertical lines to give each x-coordinate and y-coordinate concrete values.

There are two main forms of a linear equation. Let's take the equation graphing linear equations: y=2x+1 formula as an example. This equation is already in the slope-intercept form:

graphing linear equations: Example of a slope-intercept form

graphing linear equations: Example of a slope-intercept form

It is called the slope-intercept form because it easily tells you the y-intercept and the slope, or steepness of the line, from the equation.

graphing linear equations: Slope-intercept form in a graph

The slope, or steepness of the line, is the value m. The y-intercept is the value of b.

To graph it, first find the y-intercept on the graph. For this example, it intersects the y-axis at (0, 1). Then, look at the coefficient in front of x. This is the slope. You can continue drawing the line up or down depending on the slope of the line. In this case, the slope is 2, which means the line goes up the y-axis twice as fast as it goes right along the x-axis.

The second form is called point-slope form:

Example of a point-slope form

You can easily put the graph of a linear equation into this form if you already know a starting point as well as a slope. Just substitute the coordinates for  x1 and y1 formula like so:

Example of a point-slope form

Notice how with a little bit of algebra, we get it back into standard or slope-intercept form.

Example of a point-slope form

Example of a point-slope form

Graphing Linear Equations

There are two main forms for a linear equation, slope-intercept and point-slope form. Both forms include a value for the slope. The slope-intercept form tells you where the line intersects the y-axis, while the point-slope form allows you to start from any point already existing on the line.

Once you recognize these two forms, you will be able to more confidently translate between graph form and equation form.

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