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Algebra 1 Systems of Equations and Inequalities STUDY GUIDE
Name: ___________________________________________ Date: _________________ Block: _________
Algebra 1 Systems of Equations and Inequalities STUDY GUIDE
SOLs: A.4, A.5
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 ( x, y) 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 yintercepts are different.
o Infinitely many solutions: Lines coincide in the coordinate plane. Slopes and yintercepts 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.
Algebra 1 Systems of Equations and Inequalities STUDY GUIDE Page 2
Study Questions
1) For each of the following, is the ordered pair a solution to the system of equations?
a) (3, -1)
b) (1, 0)
c) (-3, -2)
d) (2, 8)
2x  2 y  4
5 x  y  16
7x  3y  7
4 y  4x  4
2 x  6 y  18
8 x  y  22
x  y6
2 y  5x  6
2) Solve following the system of equations graphically and algebraically (using substitution
or elimination). Show all work.
1
x2
a)
2
2x  2 y  2
5
y   x5
b)
2
5 x  2 y  10
y
c)
3x  6 y  18
2y  x  2
d)
7x  2 y  8
14 x  4 y  4
3) Solve following the system of equations algebraically (using substitution or elimination).
Show all work.
a)
2 x  2 y  6
3x  5 y  5
b)
x  2 y 1
3x  y  23
c)
x  3y  4
y  3x  12
Algebra 1 Systems of Equations and Inequalities STUDY GUIDE Page 3
4) Determine the apparent solution. If applicable, write “no solution” or “infinitely many
solutions”.
a)
b)
c)
5) Use a system of equations to answer the question. Show all work. You will not receive
full credit unless you show how you arrived at your answer.
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 $2, how many of each were sold?
6) Use a system of equations to answer the question. Show all work. You will not receive
full credit unless you show how you arrived at your answer.
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?
7) Use a system of equations to answer the question. Show all work. You will not receive
full credit unless you show how you arrived at your answer.
Yesterday at Megan’s Magazine Stand, she sold 20 newspapers and 8 magazines for
$62.00. So far today, she’s sold 4 newspapers and 2 magazines for $14. What is the
cost of a magazine?
Algebra 1 Systems of Equations and Inequalities STUDY GUIDE Page 4
8) Find the value of m, n, and p.
3m  2n  11
m  3n  11
p  m9
9) Graph and solve the system of inequalities.
1
y   x3
a)
2
y  2x  4
3
x5
b)
2
2 y  4x  6
y  x 1
c) y  1
x  3
3
1
4
d) 2 y  4 x  10
y2
y
y
Algebra 1 Systems of Equations and Inequalities STUDY GUIDE
Study Guide Answers
1) a) yes
b) no
2) a) (-2, -3)
c) no
d) yes
b) Infinitely
many solutions
c) (4, 1)
d) No solution
3) a) (-10, -7) b) (-9, -4) c) (-4, 0)
4) a) (3, 5) b) Infinitely many solutions
c) No solution
5) 300 fast tickets, 100 slow tickets
6) Cannot determine because there are
infinitely many solutions (they are the same
line)
10 x  5 y  55 where x=value of green
System:
and y=value of red
8 x  4 y  44
System:
x  y  400
3x  2 y  1100
where x=fast tickets
and y=slow tickets
7) Magazine = $4 (Newspaper = $1.50)
System:
where x=price of
20 x  8 y  62
newspaper and
y=price of magazine
4 x  2 y  14
8) m = 1, n = 4, p = 10
9) a)
b)
c)
d)
Algebra 1 Systems of Equations and Inequalities STUDY GUIDE Page 6