martin-gay, beginning algebra, 5ed 22 linear equation in two variables a linear equation in two...

Martin-Gay, Beginning Algebra, 5ed 22

Linear Equation in Two Variables

A linear equation in two variables is an equation that can be written in the form

Ax + By = C

where A, B, and C are real numbers and A and B not both 0. The graph of a linear equation in two variables is a straight line.

The form Ax + By = C is called standard form.


Graph the linear equation 2x – y = – 4.

Let x = 1.

2(1) – y = – 4 Replace x with 1.

2 – y = – 4 Simplify the left side.

– y = – 4 – 2 = – 6 Subtract 2 from both sides.

y = 6 Multiply both sides by – 1.

One solution to the equation is (1, 6).

Example



For the second solution, let y = 4.

2x – 4 = – 4 Replace y with 4.

2x = – 4 + 4 Add 4 to both sides.

2x = 0 Simplify the right side.

x = 0 Divide both sides by 2.

The second solution is (0, 4).

Example continued:



For the third solution, let x = – 3.

2(– 3) – y = – 4 Replace x with – 3.

– 6 – y = – 4 Simplify the left side.

– y = – 4 + 6 = 2 Add 6 to both sides.

y = – 2 Multiply both sides by – 1.

The third solution is (– 3, – 2).

Example continued:


Now plot all three of the solutions (1, 6), (0, 4) and (– 3, – 2).

x

y

(1, 6) (0, 4)

(– 3, – 2)Draw the line with arrows that contains the three points.

Example continued:


Since all points on the x-axis have a y-coordinate of 0, to find x-intercept, let y = 0 and solve for x

Since all points on the y-axis have an x-coordinate of 0, to find y-intercept, let x = 0 and solve for y


Example Graph y = 2

SolutionWriting in slope-intercept form: y = 0 • x + 2. No matter what number we choose for x, we find that y must equal 2.

Choose any number for x

y must always be 2

x y (x, y)

0 2 (0, 2)

4 2 (4, 2)

4 2 (4 , 2)

y = 2


Graph y = 2

When we plot the ordered pairs (0, 2), (4, 2) and (4, 2) and connect the points, we obtain a horizontal line.

Any ordered pair of the form (x, 2) is a solution, so the line is parallel to the x-axis with y-intercept (0, 2).


x y (x, y)

2 4 (2, 4)

2 1 (2, 1)

2 4 (2, 4)

x must be 2

Example Graph x = 2

SolutionWe regard the equation x = 2 as x + 0 • y = 2. We make up a table with all 2 in the x-column.

Any number can be used for y

x = 2


Graph x = 2

When we plot the ordered pairs (2, 4), (2, 1), and (2, 4) and connect them, we obtain a vertical line.

Any ordered pair of the form (2, y) is a solution. The line is parallel to the y-axis with x-intercept (2, 0).

martin-gay, beginning algebra, 5ed 22 linear equation in two variables a linear equation in two...

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