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a x + b y = c

is said to be in general form .

We must stipulate that a and b cannot both equal zero at the same time, for if they were we would have

0 x + 0 y = c

0 = c

This statement is true only if c = 0 . If c were to be any other number, we would get a false statement.

Now, we have the following:

The graphing of all ordered pairs that solve a linear equation in two variables produces a straight line.

This implies,

The graph of a linear equation in two variables is a straight line.

From these statements we can conclude,

If an ordered pair is a solution to a linear equations in two variables, then it lies on the graph of the equation.

Also,

Any point (ordered pairs) that lies on the graph of a linear equation in two variables is a solution to that equation.

The intercept method of graphing

When we want to graph a linear equation, it is certainly impractical to graph infinitely many points. Since a straight line is determined by only two points, we need only find two solutions to the equation (although a third point is helpful as a check).

Intercepts

When a linear equation in two variables is given in general from, a x + b y = c , often the two most convenient points (solutions) to fine are called the Intercepts: these are the points at which the line intercepts the coordinate axes. Of course, a horizontal or vertical line intercepts only one axis, so this method does not apply. Horizontal and vertical lines are easily recognized as they contain only one variable. (See Sample Set C .)

A graph of a line sloped down and to the right. The line intersects the x axis at a positive value of x, and the y axis at a positive value of y. The points where the line intersects the axes are labeled x-intercept and y-intercept respectively.

y -Intercept

The point at which the line crosses the y -axis is called the y -intercept . The x -value at this point is zero (since the point is neither to the left nor right of the origin).

x -Intercept

The point at which the line crosses the x -axis is called the x -intercept  and the y -value at that point is zero. The y -intercept can be found by substituting the value 0 for x into the equation and solving for y . The x -intercept can be found by substituting the value 0 for y into the equation and solving for x .

Intercept method

Since we are graphing an equation by finding the intercepts, we call this method the intercept method

Sample set a

Graph the following equations using the intercept method.

y 2 x = 3

To find the y -intercept , let x = 0 and y = b .

b 2 ( 0 ) = 3 b 0 = 3 b = 3

Thus, we have the point ( 0 , 3 ) . So, if x = 0 , y = b = 3 .

To find the x -intercept , let y = 0 and x = a .

0 2 a = 3 2 a = 3 Divide by -2 . a = 3 2 a = 3 2

Thus, we have the point ( 3 2 , 0 ) . So, if x = a = 3 2 , y = 0 .

Construct a coordinate system, plot these two points, and draw a line through them. Keep in mind that every point on this line is a solution to the equation y 2 x = 3 .

A graph of a line passing through two points with coordinates zero, negative three and three over two, zero.

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2 x + 3 y = 3

To find the y -intercept , let x = 0 and y = b .

2 ( 0 ) + 3 b = 3 0 + 3 b = 3 3 b = 3 b = 1

Thus, we have the point ( 0 , 1 ) . So, if x = 0 , y = b = 1 .

To find the x -intercept , let y = 0 and x = a .

2 a + 3 ( 0 ) = 3 2 a + 0 = 3 2 a = 3 a = 3 2 a = 3 2

Thus, we have the point ( 3 2 , 0 ) . So, if x = a = 3 2 , y = 0 .

Construct a coordinate system, plot these two points, and draw a line through them. Keep in mind that all the solutions to the equation 2 x + 3 y = 3 are precisely on this line.

A graph of a line passing through two points with coordinates zero, one and negative three over two, zero.

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4 x + y = 5

To find the y -intercept , let x = 0 and y = b .

4 ( 0 ) + b = 5 0 + b = 5 b = 5

Thus, we have the point ( 0 , 5 ) . So, if x = 0 , y = b = 5 .

To find the x -intercept , let y = 0 and x = a .

4 a + 0 = 5 4 a = 5 a = 5 4

Thus, we have the point ( 5 4 , 0 ) . So, if x = a = 5 4 , y = 0 .

Construct a coordinate system, plot these two points, and draw a line through them.

A graph of a line passing through two points with coordinates zero, five and five over four, zero.

Got questions? Get instant answers now!

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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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