Chapter 2 Lesson 1 Algebraic and Graphical Solutions of Linear Equations

Examples

Algebraic Solution of Linear Equations Steps for solving a linear equation in one variable 1.If a linear equation contains fractions, multiply both sides of the equation by a number that will remove all denominators from the equation. If there are two or more fractions, use the least common denominator (LCD) of the fractions. 2.Remove any parentheses or other symbols of grouping. 3.Perform any additions or subtractions to get all terms containing the variable on one side (usually left) and all other terms on the other side of the equation. Combine like terms. 4.Divide both sides of the equation by the coefficient of the variable. 5.Check the solution by substitution in the original equation. If a real- world solution is desired, check the algebraic solution for reasonableness in the real-world situation.

More Examples Mr. Immerman is paying off a credit card debt. The interest paid on a $10,000 debt over 3 years is approximated by y = x – when the interest rate is x%. What is the interest rate if the interest is $ ?

Even More Examples (Stock Market) For a period of time, a man is very successful speculating on an Internet stock, with its value growing to $100,000. However, the stock value drops rapidly until its value is reduced by 40%. What percent increase will have to occur before the latest value returns to $100,000?

Zero of a Function Any number a for which f(a)=0 is called a zero of the function f(x). If a is real, a is an x-intercept of the graph of the function. The following three concepts are numerically the same: 1.The x-intercepts of the graph of y = f(x) 2.The real zeros of the function f 3.The real solutions to the equation f(x) = 0

Examples For the function f(x) = 13x – 39, find: A) f(3) B) The zero of f(x) = 13x – 39 C) The x-intercept of the graph of y = 13x-39 D) The solution to the equation 13x – 39 = 0

Graphical Solution of Linear Equations Rewrite equation so that everything is on the right side of the equal sign Follow steps from Lesson 1-4 to find x-intercept

Other Graphical Solution of Linear Equations On your calculator, in y= menu Plug in both sides of equal sign in their own equation Press GRAPH Press 2 nd, TRACE Select option 5: intersect Select both lines Press ENTER Intersection must be in viewing window to work Solution is x value on bottom left of screen Called intersection method

Example

Real World Example The function y=0.55x describes the mean time y served in prison for a crime as a function of the mean sentence length x, where x and y are each measured in months. To find the sentence for a crime that would give an expected time served of 10 years, write an equation and solve it by using a)the x-intercept method, and b) the intersection method

Literal Equations An equation with at least 2 variables.

Simple Interest A = P(1 + rt), solve for r

Solving for Y

Direct Variation

Direct Variation Example Does the circumference of a circle vary directly with the radius of the circle? C=2πr

Homework Pages ,3,5,7,9,11,13,15,17-21,23,29,33,35,43, 53,55,60,62,66,69