Systems of Equations and Inequalities

1. Systems of Equations and Inequalities Chapter 7

2. 7-1 Solving Systems by Graphing • Combining two or more equations together (usually joined by set brackets) forms a system of equations • The solution to a system of equations is the point where they all intersect • The solution is the point that you can plug into all of the equations and it makes all of them true • Ex1. Is (4, -3) a solution of • One way to solve a system of equations is to graph all of the equations and determine the intersection point • This way is only effective if the answers are integers

3. One question per graph • Solve each system by graphing • Ex2. • Ex3.

4. If the linear equations are parallel, there will be no solution • Write “no solution” as your answer • If the linear equations are the same line, there will be infinitely many solutions • Write “infinitely many solutions” as you answer • See the bottom of page 342 for graphic depictions

5. Graphing Calculators and Systems of Equations • If you input both equations (in slope-intercept form) into a graphing calculator, it will give you the solution • Press Y= • Input one equation in Y1 and the other in Y2 • Press GRAPH • Press 2nd and then TRACE (now CALC) • Choose intersect (choice 5) • Your blinky light will be on one of the lines (called first curve), so hit ENTER • The blinky light will jump to the other line (called second curve), so hit ENTER

6. Now you need to guess where the intersection is, so use the left or right arrow keys to put the blinky light near the intersection and press ENTER • The solution is at the bottom of your screen • Use the graphing calculator to find the solution to each of the following systems • Ex1. • Ex2.

7. 7-2 Solving Systems Using Substitution • Another method for solving systems of equations is substitution • This method is most useful when the solution is not an integer and at least one of the equations is in slope-intercept form • With substitution you are substituting an equivalent expression for one of the variables • Solve one of the equations for one of the variables • It does not matter which one you choose, you will get the same answer, so choose the simplest one • This step may already be done for you

8. Plug the expression you found into the OTHER equation for that variable • Now you have a multi-step equation to solve • You MUST show all of the appropriate work • Solve each system of equations using substitution • Ex1. Ex2. • Ex3.

9. 7-3 Solving Systems Using Elimination • Elimination is another way to solve systems of equations • It is best used when one or both of the equations are in standard form and/or the answer is not integers • 1) Write the equations in standard form (align the variables) • 2) Multiply through one or both of the equations so that the coefficients for one of the variables (it doesn’t matter which one) are opposites • This step may be done for you at times • 3) Add the like terms (straight down the columns), eliminating one of the variables • 4) Solve for both variables

10. Solve each system using elimination • Ex1. • Ex2. • Ex3.

11. 7-4 Applications of Linear Systems • Systems frequently occur in real-world situations • You will have to define your variables, write your system of equations, determine which method is best for solving, and then solve (showing all of the appropriate work) • Ex1. A chemist has one solution that is 50% acid. She has another solution that is 25% acid. How many liters of each type of acid solution should she combine to get 10 liters of 40% acid solution? • Ex2. Suppose you have a typing service. You buy a personal computer for $1750 on which to do your typing. You charge $5.50 per page for typing. Expenses are $0.50 per page for ink, paper, electricity, etc. How many pages must you type to break even?

12. 7-5 Linear Inequalities • Just like graphing inequalities on a number line, graphing inequalities on a coordinate plane will require shading • The shading indicates all of the possible solutions • On a number line < and > mean to use an open circle • On a coordinate plane < and > mean to use dashed lines • On a number line < and > mean to use a closed circle • On a coordinate plane < and > mean to use solid lines • If the inequality is in slope-intercept form, then < and < mean to shade below the line • If the inequality is in slope-intercept form, then > and > mean to shade above the line • Be sure you can still tell if the line is solid or dashed!

13. If the inequality is in standard form, you have two options to determine on which side to shade • 1) write the inequality in slope-intercept form and follow the rules given • 2) pick a test point that is not on the line and see if the inequality is true at that point • Shade on the side of the line which is true • You are only allowed one graph per coordinate plane when dealing with inequalities • Ex1. Is (3, -6) a solution to y < -3x + 5?

14. Graph each inequality on its own graph • Ex2. • Ex3. 3x – 4y > 12 • Ex4. 8x + 6y > 24

15. 7-6 Systems of Linear Inequalities • Combining two or more linear inequalities creates a system of linear inequalities • The solution is the area where all of the inequalities are true (where the shading overlaps) • Do all of the original shading VERY lightly and then darken in the final answer • For a test point to be a solution to the system, it must work for all of the inequalities • One question per coordinate grid • Ex1. Is (3, 5) a solution to the following system?

16. Solve each system by graphing • Ex2. • Ex3.

17. Ex3. Suppose you have two jobs, babysitting, which pays $5 per hour, and sacking groceries, which pays $6 per hour. You can work no more than 20 hours each work, but you need to earn at least $90 per week. How many hours can you work at each job? Write a system of inequalities and graph.