Name: Date: Study Guide: Systems of Equations and Inequalities

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1 Name: Date: Study Guide: Systems of Equations and Inequalities Systems of Equations Linear systems consist of two or more linear equations in the same variables. A solution to the linear system of equations is an ordered pair ( xy, ) that is a solution to each equation in the system. Solutions to linear equations occur where the lines intersect. Methods to solve systems of linear equations: o Graphing: Graph both equations in the same coordinate plane. The point where the lines intersect is the solution. Check your estimated point by substituting the ordered pair into both of the original equations. o Substitution: Solve one of the equations for one of its variables. Substitute the expression from first step into the other equation and solve for the other variable. Substitute the value from second step into the revised equation from step 1 and solve. Verify your ordered pair solution. o Elimination: Multiply through one or both equations by a constant so that a variable will drop out if equations are added or subtracted. Add or subtract the equation to eliminate one variable. Solve the resulting equation for the other variable. Substitute in one of the original equations to find the value of the eliminated variable. Verify your ordered pair solution. Determining number of solutions: o One solution: Lines intersect in the coordinate plane. Slopes are different o No solution: Lines are parallel and never intersect. Slopes are the same but y-intercepts are different. o Infinitely many solutions: Lines coincide in the coordinate plane. Slopes and y-intercepts are the same. Systems of Inequalities System of linear inequalities consist of two or more linear inequalities in the same variable Solutions to systems of linear inequalities are ordered pairs that are a solution to each inequality in the system. o Graph to show all solutions to systems of linear inequalities. o Graphing method: graph each inequality in the same plane (for or, use a solid line; for < or > use a dashed line). Shade the half-plane that contains solutions to each inequality. Find the intersection of each of the half plane solutions. These are the solutions to the inequality.

2 Solving Systems of Equations 1) For each of the following, is the ordered pair a solution to the system of equations that it is directly above? a) (3, -1) b) (1, 0) c) (-3, -) d) (, 8) x y 4 5x y 16 7x 3y 7 4y 4x 4 x 6y 18 8x y x y 6 y 5x 6 ) Solve following the system of equations graphically and algebraically (using substitution or elimination). Show all work. a) 1 y x x y b) 5 y x 5 5x y 10 c) 3x 6y 18 y x d) 7 x y 8 14x 4y 4 3) Determine the apparent solution. If applicable, write no solution or infinitely many solutions. a) b) c)

3 Use a system of equations to answer the question. Show all work. 4) At Funland Amusement Park, rides are categorized as Fast and Slow. On one Tuesday, the park sold a total of 400 tickets for $1100. If the price of a Fast ticket is $3 and the price of a Slow ticket is $, how many of each were sold? 5) Hailey and Olivia are playing the board game Kurbopple, where you can earn coins if you roll a green or a red. Over the course of the game, Hailey rolled 10 greens and 5 reds and earned 55 coins. Olivia rolled 8 greens and 4 reds and earned 44 coins. How much is a red roll worth? How much is a green roll worth? 6) Yesterday at Megan s Magazine Stand, she sold 0 newspapers and 8 magazines for $6.00. So far today, she s sold 4 newspapers and magazines for $14. What is the cost of a magazine?

4 Linear Inequalities 7) When would the boundary line of the graph of a linear inequality be dashed? When would it be solid? 8) How do you know whether to shade above or below the line when graphing an inequality on the coordinate plane? 9) Which describes the solution of the inequality y >? A solid vertical line through (0, ) with shading to left of line B C D dashed vertical line through (0, ) with shading to left of line solid horizontal line through (0, ) with shading below line dashed horizontal line through (0, ) with shading above line 10) Which ordered pair is a solution of the inequality 4x + 5y < 1? a. (-, -6) b. (11, -1) c. (5, 4) d. (4, 5) 11) Use the linear inequality 4x y < 6 for parts a-c. a. Write the inequality in slope intercept for b. Graph the inequality on the grid to the right. c. Name at least 3 possible solutions to this inequality. Explain how you know that these ordered pairs are solutions to the inequality.

5 1) Carlos has at most $0 to spend on a bouquet of flowers. Carnations cost $1 each and roses $ each. a. Write and graph the inequality that shows the number of carnations and roses Carlos can buy. b. Give an example of a possible combination of flowers Carlos can buy within his budget. Write your answer in a complete sentence. Justify your answer using your graph.

6 Systems of Inequalities 13) Write a system of inequalities for the graphs given. Give two possible solutions to each system. Then, plug in your solutions in the system that you wrote to verify that you are correct. A. B. System: System: Solutions: Solutions: (, ) & (, ) (, ) &(, ) 14) Graph and solve the system of inequalities. a. 1 y x 3 y x 4 b. 3 y x 5 y 4x 6

7 15) Larissa plans to bake at most 10 loaves of bread. She makes x loaves of banana bread that sell for $1.5 each and y loaves of nut bread that sell for $1.50 each. She hopes to make at least $4 in sales. Write and graph a system of inequalities for this situation. What does the graph show? Give a possible combination that would allow Larissa to meet her goal.