Name: ________________________________________ Date: ___________________________ Hour: ______________

Unit 4 Review Key 1. Circle the point or points below that represent the solution(s) to the system of equations graphed to the

right, and then explain below how many solutions there are and how you know that those are the correct solutions. This system has three solutions. I know this because the solution to a system of equations is the point (or points) that makes both equations true. That will be the point (or points) where the graphs cross each other. This system has three intersections, so the system has three solutions.

2. The graph to the left shows two linear equations. How many solutions does this system have? Explain

your reasoning. 3. This system will have on solution. I know this because the equations are linear and their slopes are not the same so they are not parallel. This means they will cross and have one solution.

4. Add in a line to complete the systems of equations below. Sketch your line so that the system will have

no solutions, then write the equation of the line you sketched. Systems of equations that have no solutions are parallel lines because they never cross. So we need to sketch in a line parallel to the one that is drawn. In order to make it parallel they have to have the SAME SLOPE. Slope is rise/run, so count the rise and the run from the two points shown (they cross nicely). So the slope is

Rise=4

Run=5

4/5. The equation should be y=mx+b. My equation needs to have a slope (m) of 4/5 and then the y-intercept (b) can be whatever it wants to be.

Y=4/5x+1 would work. Y=4/5x+anything would work.

5. Construct a system of two linear equations where (−3,2) is a solution to the first equation but not to the

second equation, and where (4,−3) is a solution to your system. Graph your system, and then explain how your graph shows that your system satisfies the required conditions.

6. These two equations form a system of equations:

• The equation 3𝑥 + 2𝑦 = −4 • The equation of the line that passes through the points (0, 3) and (2, 7). Does this system have a solution? Explain how you know. In order to solve this we will need to find the equation that goes through the points (0,3) and (2,7). The point (0,3) tells me that 3 has to be the b because it’s the y-value when the x-value is 0. So now I just need the slope. It’s change in y over change in x, so it will be 3-7/0-2=-4/-2=2. So the equation y=mx+b is y=2x+3. Now you can solve the system however you want. I would graph the two equations and see that they cross. That means that the system has a solution. If you use elimination or substitution to solve this you will need to find the solution in order to prove that there is one. You could also use your calculator to solve the system and show that there is a solution.

7. Solve the system.

𝑦 = −2𝑥 + 3 3𝑥 + 5𝑦 = 15 Solve this using any method you would like. I would probably graph them and see where they cross, or use the calculator to solve the system by pressing Menu, Algebra, Systems of Linear Equations. The answer is (0,3). 8. A farmhouse shelters 10 animals. Some are pigs and some are ducks. There are 36 legs altogether. How

many of each animal are there? My two variables are the number of pigs (p) and the number of ducks (d). My equations would be p+d=10 because there are 10 animals total, and 4p+2d=36 because there are 4 legs per pig and 2 legs per duck and the legs add up to 36 legs total.

Solve this system using any method. I graphed it and found the intersection. The solution, assuming p=x and d=y, would be (8,2). So there are 8 pigs and 2 ducks. 9. You are going to rent a bowling alley for your birthday party. There are two bowling alley packages to

choose from. The first package costs $150 to rent the lanes and then you have to pay an additional $2 for each person that comes to the party. The second package only costs $50 for the lanes, but you must pay $4 for each person that comes to the party. How many people will you have to invite in order for the two plans to cost exactly the same amount? The equations are linear because in both cases the cost for the party is growing by a constant amount. The two equations will be y=2x+150 and y=4x+50. Solve this system however you want. I would graph in on my calculator and have my calculator find the intersection. The solution is (50,250) so we would need to invite 50 people in order for the parties to cost the same amount (and both parties would cost $250).

10. Brenda’s school is selling tickets to a spring musical. On the first day of ticket sales the school sold 3

senior citizen tickets and 9 child tickets for a total of $75. The school took in $67 on the second day by selling 8 senior citizen tickets and 5 child tickets. What is the price each of one senior citizen ticket and one child ticket? My two variables are the senior ticket price (s) and the child ticket price (c). My equations would be 3s+9c=75 8s+5c=67 Solve this using any method you want. I would graph it and find the intersection on the calculator. The solution, assuming s=x and c=y, would be (4,7). So the senior ticket is $4 and the child ticket is $7. 11. Write the system of inequalities modeled by the graph. Let’s start by writing the equations of the lines. Y=mx+b. m=rise/run, b=y-intercept. The top line has a y-intercept of 5 and a rise/run of ½.The bottom line has a y-intercept of -2 and a rise/run of -1/1=-1. 1 𝑦 = 𝑥+5 2 𝑦 = −𝑥 − 2 but we need them to be inequalities because they are shaded. The way the inequality is facing determines which part is shaded. The shaded part is the double shaded part, which means we must have shaded below both lines. (0,0) is a solution for the solid line (because (0,0) is below the line) but (0,0) is NOT a solution for the dotted line (because (0,0) is above the line). 1

0____ (0) + 5 2

0 ____5 we need to put in the inequality that makes this

TRUE because

1

(0,0) is a solution for this line. So 𝑦 < 𝑥 + 5 2

0____ − (0) − 2 0 ____ − 2 we need to put in the inequality that makes this FALSE because (0,0) is NOT solution for this line. So 𝑦 < −𝑥 − 2 The top line is solid, so it gets an equal under the inequality. The bottom line is dotted, so it does not get an equal under the inequality.

1 𝑦≤ 𝑥+5 2 𝑦 < −𝑥 − 2

12. Find the value of two numbers if their sum is 38 and their difference is 8.

You can guess and check on this if you want. Sum means add and difference means subtract, so we need two numbers that add to 38 and also subtract to 8. You can also treat it like a system of equations if we say the two numbers are x and y. x+y=38 and x-y=8 Solve that however you want. I would type it into the calculator by pressing menu, algebra, system of linear equations solve. 13. Solve the system.

3𝑦 + 2𝑥 = 6 5𝑦 − 2𝑥 = 10 Solve this using any method you would like. I would probably graph them and see where they cross, or use the calculator to solve the system by pressing Menu, Algebra, Systems of Linear Equations. The answer is (0,2). 14. Graph the system of inequalities.

2𝑥 + 𝑦 < 3 −3𝑥 + 𝑦 ≥ 9 You can type this into your calculator to graph it and then copy the graph onto your test. Or you can graph it by hand, but then they would need to be in y=mx+b. Get them there by subtracting the x’s. So the equations will be: 𝑦 < −2𝑥 + 3 𝑦 ≥ 3𝑥 + 9 The b tells where it crosses the y-axis and the m tells the rise/run. Then we test a point and shade. I will test (0,0). 0 < −2(0) + 3 0 ≥ 3(0) + 9

0 < 3 𝑇𝑅𝑈𝐸, 𝑠𝑜 𝑠ℎ𝑎𝑑𝑒 𝑠𝑖𝑑𝑒 𝑤𝑖𝑡ℎ (0,0) 0 ≥ 9 𝐹𝐴𝐿𝑆𝐸, 𝑠𝑜 𝑠ℎ𝑎𝑑𝑒 𝑠𝑖𝑑𝑒 𝑤𝑖𝑡ℎ𝑜𝑢𝑡 (0,0)

15. Solve the system. 5

𝑦 = −3𝑥 + 3 1

𝑦 = 3𝑥 − 3 Solve this using any method you would like. I would probably graph them and see where they cross, or use the calculator to solve the system by pressing Menu, Algebra, Systems of Linear Equations. The answer is (3,-2).

16. Solve the system.

6𝑥 − 12𝑦 = 24 −𝑥 − 6𝑦 = 4 Solve this using any method you would like. I would probably graph them and see where they cross, or use the calculator to solve the system by pressing Menu, Algebra, Systems of Linear Equations. The answer is (2,-1).