Chapter P.4 Review Group E

Solving Equations Algebraically and Graphically When solving equations identify these points: - Conditional: Sometimes true, Finite number of answers - Identity: Always true, Infinite number of answers Domain Restrictions: Values that make the function invalid. Domain restrictions occur most often with denominators equaling zero or radicals with even radicands being negative.

Linear Equations A linear equation in one variable (x) is an equation that can be written in the standard form A(x)+B=0 with A not equaling 0 Solving Equations involving Fractions To solve an equation involving fractional equations, find the Least Common Denominator of all terms in the equation and multiply every term by the LCD, Which clears equation of fractions Example: (x/3)+(3x/4)=2 (12)(x/3)+(12)(3x/4)=(12)(2) 4x+9x=24 13x=24 x=24/13

Graphical Approximations of Solutions of an equation 1. Write the equation in general for y=0 with the nonzero terms on one side of the equation and zero on the other side 2. Use a graphing utility to graph the equation. Be sure the viewing window shows all the relevant features of the graph 3. Use the zero or root feature or the zoom and trace feature of the graphing utility to approximate each of the x-intercepts of the graph. Remember that a graph can have more than one x-intercepts so you need to change the viewing window a few times

Definitions of Intercepts The point (a, 0) is called an x-intercept of the graph of an equation if it is a solution point. To find let y=0 and solve equation for x The point (0,b) is called y-intercept of the graph of an equation if it is a solution point. To find let x=0 solve the equation for y Example: Find x- and y- intercepts of the graph of 2x+3y=5. Let y=0 to find x-intercepts. 2x=5 to x=5/2. Let x=0 to find y-intercept, 3y=5 to y=5/3 An equation involving a radical expression can often be cleared of radicals by raising both sides of the equation to an appropriate power

Points of intersection of two graphs To find points of intersection of the graph of two equations, solve each equation for y (or x) and set the two results equal to each other. The resulting equation will be an equation in one variable, which can be solved using standard procedures. Algebraic: solve equation for y Y=2/3x+2/3 and y=4x-6 Set two equations for y equal to each other and solve resulting equation for x 2/3x+2/3=4x-6 (3)(2/3x)+(3)(2/3)=(3)(4x)-(3)(6) 2x+2=12x x=-20 x=2 When x=2 the y value of each of the given equation is 2 so the point of intersection is (2,2)

Points of intersection of two graphs (Cont.) Graphical Solution: Solve each equation for y to obtain y=2/3x+2/3 and y=4x-6. Then use the calculator to graph both equations in the same window, use the intersect feature of the graphing utility to approximate the point of intersection to be (2,2)