Systems of equations Vision 1 Prepared September 2010 by C. Cichanowicz

y = ax + b (function form) Equation of a Line y = ax + b (function form) Graphing: No matter what form the equation is given in (function or general form), manipulate the equation so that it is in this form a – the rate of change (or slope in the graph) b – the initial value Prepared September 2010 by C. Cichanowicz

Equation of a Line The slope In a graph: it is the steepness of the line (if the line goes down, the slope is negative...if the line goes up, the slope is positive) In a word problem: it is the value at which something increases or decreases. (Ex. A salary of 9$/hr, decreases by 10% daily) Prepared September 2010 by C. Cichanowicz

Equation of a Line Slope (rate of change): Prepared September 2010 by C. Cichanowicz

Equation of a Line Determine the slope Prepared September 2010 by C. Cichanowicz

Equation of a Line The initial value In a graph: it is where the line crosses the y-axis In a word problem: it is the starting value before the rate is taken into consideration. Ex. A base salary before commission An employee makes 200$ plus 50$/computer sold Initial value Slope or rate of change Prepared September 2010 by C. Cichanowicz

Equation of a Line Determine the equation of the line from the graph Prepared September 2010 by C. Cichanowicz

Equation of a Line Determine the equation of the line from the graph Prepared September 2010 by C. Cichanowicz

Equation of a Line Graphing equations From the equation of the line determine the initial value, this will be your starting point on the y-axis From the equation of the line establish the value of the slope If given as a fraction, remember: If given as a whole number, the x-value is 1 + go up - go down + go right - go left Prepared September 2010 by C. Cichanowicz

Equation of a Line The form: Ax + By + C = 0 When an equation is given in this form transform it into the form y = ax + b 4x + 3y – 15 = 0 Isolate y (any operations you perform will be performed on each side of the equation 4x – 4x + 3y – 15 + 15 = - 4x + 15 3y = - 4x + 15 Divide by the coefficient of y Simplify: Prepared September 2010 by C. Cichanowicz

Systems of Equations Word problems can be turned into a system of equations... In the form : y=ax+b In the form: ax+by=c (this is usually some kind of cost equation) Solving Systems: to determine intersection coordinates Substitution Comparison Elimination Prepared September 2010 by C. Cichanowicz

Systems of Equations The solution is (19, -25) Substitution 2x + y = 13 4x + 3y = 1 Transform one of the equations to function form 2x – 2x + y = 13 – 2x y = 13 – 2x Substitute the transformed equation into the other equation 4x + 3 (13 – 2x) = 1 4x + 39 – 6x = 1 -2x + 39 = 1 -2x + 39 - 39 = 1 -39 -2x = -38 -2x ÷ -2 = -38 ÷ -2 x = 19 To find y, substitute the value obtained for x into the transformed equation y = 13 – 2x y = 13 – 2(19) y = 13 – 38 y = -25 The solution is (19, -25) Prepared September 2010 by C. Cichanowicz

Systems of Equations Comparison The Solution is: (-2, 5) y = 2x + 9 Both equations are equal to y, we can therefore say they are equal to each other (solve by algebraic manipulation) 2x + 9 = -3x – 1 2x + 3x + 9 = -3x + 3x – 1 5x + 9 = -1 5x + 9 – 9 = -1 – 9 5x = -10 5x ÷ 5 = -10 ÷ 5 x = -2 To find y, substitute the value obtained for x into one of the equations y = 2x + 9 y = 2(-2) + 9 y = -4 + 9 y = 5 The Solution is: (-2, 5) Prepared September 2010 by C. Cichanowicz

Systems of Equations Elimination 2x + 5y = -4 3x -2y = 13 Make one of the variables the same Subtract the 2 equations To find y, substitute the value obtained for x into one of the equations 2x + 5y = -4 2x + 5(-2) = -4 2x – 10 = -4 2x – 10 + 10 = -4 + 10 2x = 6 2x ÷ 2 = 6 ÷ 2 x = 3 The solution is: (3, -2) Prepared September 2010 by C. Cichanowicz

Systems of Equations Solve the following systems using the method of your choice 5x – 3y = 12 y = -2x + 18 2x + 3y = 6 5x + 10y = 20 y = -2x + 9 y = x - 3 Prepared September 2010 by C. Cichanowicz