Chapter 2 Section 2 Linear Equations

Linear equations Linear equations are equations where the exponent of the variables is one variables are not multiplied together -- only added and subtracted a variable does NOT appear in the denominator A graph of the solutions yields a straight line

Examples Linear equations Non-linear equations

Linear functions A linear function is any function whose graphed solution pairs are a straight line, i.e. it can be written in the form f(x) = ax + b. The variable must have an exponent of one, not appear in the denominator, and have a real coefficient.

Linear functions? State whether each function is a linear function. Explain.

Standard Form of a Linear Equation The standard form of a linear equation is: Ax + By = C, where A is a whole number B and C are integers A and B are not BOTH zero

Writing linear equations in standard form To write a linear equation in standard form put both variables on one side of the equation in alphabetical order and constants on the other make the leading coefficient positive by multiplying both sides by negative one, if necessary clear the denominators by multiplying both sides by the least common denominator reduce the equation by dividing both sides by a common factor, if there is one

Standard Form: Example 1 Write the equation in standard form. Identify A, B, and C: y = 3x - 9 -3x + y = -9 3x - y = 9 A = 3, B= -1, C = 9

Standard Form: Example 2 Write the equation in standard form. Identify A, B, and C: -(2/3)x = 2y - 1 -(2/3)x - 2y = -1 (2/3)x + 2y = 1 2x + 6y = 3 A = 2, B = 6, C = 3

Standard Form: Example 3 Write the equation in standard form. Identify A, B, and C: 8x - 6y + 4 = 0 8x - 6y = -4 4x - 3y = -2 A = 4, B = -3, C = -2

Standard Form and x- and y-intercepts Remember that the graph will cross the y-axis when x = 0, and will cross the x-axis when y = 0. This fact can be used to graph most -- though not all -- linear equations. Plug y = 0 into the equation to solve for the x-intercept (x, 0), and x = 0 in to find the y-intercept ( 0, y).

Last example Find the x- and y-intercepts of -2x + y - 4 = 0. Then graph the equation. Put in standard form: -2x + y - 4 = 0 -2x + y = 4 2x - y = -4 Make x = 0 to find y-intercept 2(0) - y = -4 -y = -4 y = 4 Make y = 0 to find x-intercept 2x - (0) = -4 2x = -4 x = -2 Graph:

Your assignment… Chapter 2 section 2, 15-24, 27-55, and 69-78. REMEMBER: Where you see f(x) -- or g(x) or h(x) --you can replace it with y to make a function into an equation (see 49 and 50). What is now proved was once only imagined. --William Blake