Understanding Linear Equations: Basics and Examples

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This content covers the basics of linear equations, including their standard form Ax + By = C and how to identify them. It also discusses examples of linear and nonlinear equations, x and y-intercepts, and how to find them. The visual aids and explanations provided make it easier to grasp these fundamental concepts in mathematics.

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08/15/2019 Agenda Number Sense Routine Cornell Notes Student/teacher dialog Number Sense Routine 5 + ( 12) 10 + ( 6) 9 + ( 13) 12 + 17

Identifying a Linear Equation Ax + By = C The exponent of each variable is 1. The variables are added or subtracted. A or B can equal zero. A > 0 Besides x and y, other commonly used variables are m and n, a and b, and r and s. There are no radicals in the equation. Every linear equation graphs as a line.

Examples of linear equations 2x + 4y =8 6y = 3 x x = 1 Equation is in Ax + By =C form Rewrite with both variables on left side x + 6y =3 B =0 x + 0 y =1 -2a + b = 5 Multiply both sides of the equation by -1 2a b = -5 4 x y = = Multiply both sides of the equation by 3 4x y =-21 7 3

Examples of Nonlinear Equations The following equations are NOT in the standard form of Ax + By =C: The exponent is 2 4x2 + y = 5 xy + x = 5 s/r + r = 3 There is a radical in the equation x = = 4 Variables are multiplied Variables are divided

x and y -intercepts The x-intercept is the point where a line crosses the x-axis. The general form of the x-intercept is (x, 0). The y-coordinate will always be zero. The y-intercept is the point where a line crosses the y-axis. The general form of the y-intercept is (0, y). The x-coordinate will always be zero.

Finding the x-intercept For the equation 2x + y = 6, we know that y must equal 0. What must x equal? Plug in 0 for y and simplify. 2x + 0 = 6 2x = 6 x = 3 So (3, 0) is the x-intercept of the line.

Finding the y-intercept For the equation 2x + y = 6, we know that x must equal 0. What must y equal? Plug in 0 for x and simplify. 2(0) + y = 6 0 + y = 6 y = 6 So (0, 6) is the y-intercept of the line.

To summarize. To find the x-intercept, plug in 0 for y. To find the y-intercept, plug in 0 for x.

Find the x and y- intercepts of x = 4y 5 y-intercept: x-intercept: Plug in x = 0 x = 4y - 5 0 = 4y - 5 5 = 4y = y 4 Plug in y = 0 x = 4y - 5 x = 4(0) - 5 x = 0 - 5 x = -5 5 (-5, 0) is the x-intercept 5 4 (0, ) is the y-intercept

Find the x and y-intercepts of g(x) = -3x 1* y-intercept x-intercept Plug in x = 0 g(x) = -3(0) - 1 g(x)= 0 - 1 g(x) = -1 Plug in y = 0 g(x) = -3x - 1 0 = -3x - 1 1 = -3x = x 3 1 3 1 (0, -1) is the y-intercept ( , 0) is the x-intercept *g(x) is the same as y

Find the x and y-intercepts of 6x - 3y =-18 x-intercept y-intercept Plug in y = 0 6x - 3y = -18 6x -3(0) = -18 6x - 0 = -18 6x = -18 x = -3 Plug in x = 0 6x -3y = -18 6(0) -3y = -18 0 - 3y = -18 -3y = -18 y = 6 (-3, 0) is the x-intercept (0, 6) is the y-intercept

Find the x and y-intercepts of x = 3 x-intercept y-intercept Plug in y = 0. A vertical line never crosses the y-axis. There is no y. Why? There is no y-intercept. x = 3 is a vertical line so x always equals 3. (3, 0) is the x-intercept. x y

Find the x and y-intercepts of y = -2 y-intercept x-intercept y = -2 is a horizontal line Plug in y = 0. y cannot = 0 because y = -2. y = -2 is a horizontal line so it never crosses the x-axis. so y always equals -2. (0,-2) is the y-intercept. x There is no x-intercept. y

Graphing Equations Example: Graph the equation -5x + y = 2 Solve for y first. -5x + y = 2 y = 5x + 2 Add 5x to both sides The equation y = 5x + 2 is in slope-intercept form, y = mx+b. The y-intercept is 2 and the slope is 5. Graph the line on the coordinate plane.

Graphing Equations Graph y = 5x + 2 x y

Graphing Equations Graph 4x - 3y = 12 Solve for y first 4x - 3y =12 -3y = -4x + 12 Divide by -3 12 -3 4 3 Subtract 4x from both sides -4 y = x + -3 Simplify y = x 4 4 3 The equation y = x - 4 is in slope-intercept form, y=mx+b. The y -intercept is -4 and the slope is . Graph the line on the coordinate plane. 4 3