martin-gay, beginning algebra, 5ed 22 linear equation in two variables a linear equation in two...
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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.
Martin-Gay, Beginning Algebra, 5ed 33
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
Martin-Gay, Beginning Algebra, 5ed 44
Graph the linear equation 2x – y = – 4.
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:
Martin-Gay, Beginning Algebra, 5ed 55
Graph the linear equation 2x – y = – 4.
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:
Martin-Gay, Beginning Algebra, 5ed 66
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:
Martin-Gay, Beginning Algebra, 5ed 77
Martin-Gay, Beginning Algebra, 5ed 88
Martin-Gay, Beginning Algebra, 5ed 99
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
Martin-Gay, Beginning Algebra, 5ed 1010
Martin-Gay, Beginning Algebra, 5ed 1111
Martin-Gay, Beginning Algebra, 5ed 1212
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
Martin-Gay, Beginning Algebra, 5ed 1313
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).
Martin-Gay, Beginning Algebra, 5ed 1414
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
Martin-Gay, Beginning Algebra, 5ed 1515
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 1616
Martin-Gay, Beginning Algebra, 5ed 1717