Math - Linear Equations

Linear Equations is an equation that represents a straight line.

What is a Linear Equation?

• Represents a straight line in a Cartesian coordinate system.
• Involves variables raised only to the first power (no exponents higher than 1).
• Uses equality sign = to relate the two expressions.

Examples

$y/2 = 8$
$2y = 2x + 8$
$10x = 50$
$7x + 4 = 3(3x - 8)$

Solving Linear Equations

2x + 3 = -5

$2x + 3 = -5\\ 2x + 3 - 3 = -5 - 3\\ 2x = -8\\ \frac{2x}{2} = \frac{-8}{2}\\ x = -4$

x + 3 = 2(x + 3) - 4

$x + 3 = 2(x + 3) - 4\\ x + 3 - 3 = 2(x + 3) - 4 - 3\\ x = 2(x + 3) - 7\\ x = 2x + 6 - 7\\ x = 2x - 1\\ 1x - 2x = 2x - 2x - 1\\ -1x = -1\\ -x = -1\\ -x \times -1 = -1 \times -1\\ x = 1$

$\frac{x + 8}{2} = 2(x + 1)$

$\frac{x + 8}{2} = 2(x + 1)\\ \frac{x + 8}{2} = 2x + 2\\ \cancel{2}\frac{(x + 8)}{\cancel{2}} = 2(2x + 2)\\ x + 8 = 2(2x + 2)\\ x + 8 = 4x + 4\\ x + 8 - 8 = 4x + 4 - 8\\ x = 4x - 4\\ 1x - 4x = -4\\ -3x = -4\\ x = \frac{-4}{-3}$

Graphing Linear Equations

The best way to graph a linear equation is to write the equation in slope-intercept form.

The slope-intercept formula

Graphing Linear Equations (youtube)
Graphing Linear Equations in Two Variables (palmbeachstate.edu)

Examples

4x - 7 = 21

$4x - 7 = 21\\ y = 4x - 7$

7x + 9 = 3(x + 2) + 1

$7x + 9 = 3(x + 2) + 1\\ y = 7x + 9\\ y = 3x + 7$

y - 5x = x + 5

$y - 5x = x + 5\\ y - 5x + 5x = 5x + 1x + 5\\ y = 6x + 5$

2y + 1 = 4x * 2

$2y + 1 = 4x * 2\\ 2y + 1 = 8x\\ 2y + 1 - 1 = 8x - 1\\ 2y = 8x - 1 \\ \frac{\cancel{2}y}{\cancel{2}} = \frac{8x - 1}{2}\\ y = 4x - 0.5$