Algebra 1: Systems of Equations and Inequalities
Warm-up
Name: _________________________
1. Given the system shown below do the following:
2𝑥 + 2𝑦 = −6
𝑥 = 15 + 5𝑦
(a) Solve this system graphically using the grid.
(b) Solve this system using substitution. Show your work.
2. Write the system of equations shown in each graph. Then, write the solution to each equation.
3. Use what you have learned about systems of equations to answer these questions.
a. Find values of a and b so that the system has infinitely many solutions. Explain how the graphs of the
equations will be related.
3𝑥 + 2𝑦 = 5
𝑎𝑥 + 4𝑦 = 𝑏
b. Find the values of a and b so that the system has no solutions. Explain how the graphs will be related.
3𝑥 + 6𝑦 = 12
𝑥 + 𝑎𝑦 = 𝑏
c. Find the values of a and b so that the system has exactly one solution. Explain how the graphs will be
related.
3𝑥 + 2𝑦 = 5
𝑎𝑥 + 4𝑦 = 𝑏
Algebra 1: Systems of Equations and Inequalities
Solving by Combination_Elimination
Name: _________________________
There is one final way that we will solve systems of equations. Systems are important because they tell us
multiple conditions that relate multiple variables or unknowns.
1) Consider the system show to the right and its solution (1,5).
(a) Show that 𝑥 = 1 and 𝑦 = 5 is a solution to the system of equations.
4𝑥 + 2𝑦 = 14
𝑥 − 𝑦 = −4
(b) Find the sum of the two equations. Is the point (1,5) a solution to this new equations? Justify your
yes/no response.
(c) Multiply both sides of the second equation by 2 to get an equivalent equation. Is the point (1,5) a
solutions to this new equation? Justify your yes/no response.
(d) Take the equation you found in (c) and add it to the first equation. What happens? How does this
allow us to now solve for the variable 𝑥? Do so, what do you find?
(e) Once you know the value of 𝑥, how can you find the value of 𝑦?
A solution to a system of equations remains a solution to that system under a variety of conditions.
2. Consider the system shown to the right:
(a) Show that the point (3,-1) is a solution to the system.
4𝑥 − 3𝑦 = 15
3𝑥 + 2𝑦 = 7
(b) The point (3,-1) will be a solution to the system shown below. How can you determine this without
substituting the point in?
𝟖𝒙 − 𝟔𝒚 = 𝟑𝟎
𝟗𝒙 + 𝟔𝒚 = 𝟐𝟏
(c) What happens when you add these two equations together? How can this let you solve for 𝑥? Find it
and find 𝑦.
3. Solve the system below using the method of elimination/combination. Show the steps in your work and
show that your answer in is fact a solution to the system
2𝑥 + 4𝑦 = 2
6𝑥 + 3𝑦 = −3
4. The point (4,-2) is a solution to the system of equations
equations would it not be a solution to?
(a) 3𝑥 + 6𝑦 = 0
(b) 2𝑥 + 2𝑦 = 12
2𝑥 + 𝑦 = 6
𝑥 + 5𝑦 = −6
(c) 2𝑥 + 10𝑦 = −12
Which of the following
(d) 𝑥 − 4𝑦 = 12
5. Consider the system show below. Solve the system two ways:
(a) Eliminate 𝑥 to solve
4𝑥 + 5𝑦 = 12
−2𝑥 + 𝑦 = 8
(b) Eliminate 𝑦 to solve
4𝑥 + 5𝑦 = 12
−2𝑥 + 𝑦 = 8
(c) Show that the point that you found in (a) and (b) is a solution to this system of equations.
6. Two numbers have the following properties. The sum of the larger and twice the smaller is equal to 13.
Twice their positive difference is equal to eight. What are the two numbers? Carefully define your variables
and create a system of equations to represent this situation. Solve the system that you created.
Algebra 1: Systems of Equations and Inequalities
Solving by Combination_Elimination HW
Name: _________________________
Solve the following systems of equations by combination/elimination.
1.
3.
5.
𝑥−𝑦 =7
𝑥+𝑦 =5
𝑥 − 𝑦 = 15
4𝑥 + 2𝑦 = 30
2𝑥 + 3𝑦 = 16
5𝑥 − 2𝑦 = 21
2.
2𝑥 + 5𝑦 = 3
-2𝑥 − 𝑦 = 5
4.
2𝑥 + 3𝑦 = 17
5𝑥 + 6𝑦 = 32
6.
6𝑥 − 7𝑦 = 25
15𝑥 + 3𝑦 = 42
7. Lilly and Rose are sisters. The sum of their ages is 19 and the positive difference of their ages is 9. Set us a
system of equations involving Lilly’s age, 𝐿, and Rosie’s age, 𝑅, assuming that Lilly is the older child. Solve
the system to find their ages.
8. Shana bought sodas and popcorn for the movies. Sodas cost $3 each and popcorn cost $4 per bag. Shana
bought 7 things from the concession, all either sodas or bags of popcorn. Shana spent a total of $26. Write a
system of equations involving the number of sodas, 𝑠, and the bags of popcorn, 𝑏. Solve the system to see how
many of each Shana bought.