Algebra II
Algebra II
NAME _____________________________________________ DATE ____________________________ PERIOD _____________
3-1 Study Guide and Intervention
Solving Systems of Equations
Solve Systems Graphically A system of equations is two or more equations with the same variables. You can solve a
system of linear equations by using a table or by graphing the equations on the same coordinate plane. If the lines
intersect, the solution is that intersection point. The following chart summarizes the possibilities for graphs of two linear
equations in two variables.
Graphs of Equations
Slopes of Lines
Classification of System
Number of Solutions
Lines intersect
Different slopes
Consistent and independent
One
Lines coincide (same line)
Same slope, same y-intercept
Consistent and dependent
Infinitely many
Lines are parallel
Same slope, different y-intercepts
Inconsistent
None
Example: Graph the system of equations and describe it as
consistent and independent, consistent and dependent, or inconsistent.
x – 3y = 6
2x – y = –3
Write each equation in slope-intercept form.
x–3y=6
y= x–2
2 x – y = –3
y = 2x + 3
The graphs intersect at (–3, –3). Since there is one
solution, the system is consistent and independent.
Exercises
Graph each system of equations and describe it as consistent and independent, consistent and dependent, or
inconsistent.
1. 3x + y = –2
6x + 2y = 10
2. x + 2y = 5
3x – 15 = –6y
3. 2x – 3y = 0
4x – 6y = 3
4. 2x – y = 3
x+ 2y = 4
5. 4x + y = –2
2x + = –1
6. 3x – y = 2
x+y=6
Chapter 3
5
Glencoe Algebra 2
NAME _____________________________________________ DATE ____________________________ PERIOD _____________
3-1 Study Guide and Intervention
Solving Systems of Equations
Solve Systems Algebraically To solve a system of linear equations by substitution, first solve for one variable in
terms of the other in one of the equations. Then substitute this expression into the other equation and simplify. To solve a
system of linear equations by elimination, add or subtract the equations to eliminate one of the variables.
Example 1: Use substitution to solve the system of equations.
Solve the first equation for y in terms of x.
2x – y = 9
First equation
–y = –2x + 9
Subtract 2x from both sides.
y = 2x – 9
Multiply both sides by –1.
Substitute the expression 2x – 9 for y into the second
equation and solve for x.
x + 3y = –6
Second equation
x + 3(2x – 9) = –6
Substitute 2x – 9 for y.
x + 6x – 27 = –6
Distributive Property
7x – 27 = –6
Simplify.
7x = 21
Add 27 to each side.
x=3
Divide each side by 7.
2x – y = 9
x + 3y = –6
Now, substitute the value 3 for x in either original
equation and solve for y.
2x – y = 9
2(3) – y = 9
6–y=9
–y = 3
y = –3
First equation
Replace x with 3.
Simplify.
Subtract 6 from each side.
Multiply each side by –1.
The solution of the system is (3, –3).
Example 2: Use the elimination method to solve the system of equations.
3x – 2y = 4
5x + 3y = –25
Multiply the first equation by 3 and the second equation
by 2. Then add the equations to eliminate the y variable.
3x – 2y = 4
Multiply by 3.
9x – 6y = 12
5x + 3y = –25 Multiply by 2. 10x + 6y = –50
19x
= –38
x
= –2
Replace x with –2 and solve for y.
3x – 2y = 4
3(–2) – 2y = 4
–6 – 2y = 4
–2y = 10
y = –5
The solution is (–2, –5)
Exercises
Solve each system of equations.
1. 3x + y = 7
2. 2x + y = 5
4x + 2y = 16
3x – 3y = 3
3. 2x + 3y = –3
x + 2y = 2
4. 2x – y = 7
6x – 3y = 14
5. 4x – y = 6
2x – 2 = 4
6. 5x + 2y = 12
–6x – 2y = –14
7. 2x + y = 8
3x + 𝑦 = 12
8. 7x + 2y = –1
4x – 3y = –13
9. 3x + 8y = –6
x–y=9
Chapter 3
6
Glencoe Algebra 2
NAME _____________________________________________ DATE ____________________________ PERIOD _____________
3-2 Study Guide and Intervention
Solving Systems of Inequalities by Graphing
Systems of Inequalities To solve a system of inequalities, graph the inequalities in the same coordinate plane.
The solution of the system is the region shaded for all of the inequalities.
Example: Solve the system of inequalities.
𝒙
y ≤ 2x 1 and y > 𝟑 + 2
The solution of y ≤ 2x – 1 is Regions 1 and 2.
The solution of y > + 2 is Regions 1 and 3.
3
The intersection of these regions is Region 1, which is
the solution set of the system of inequalities.
Exercises
Solve each system of inequalities by graphing.
1. x – y ≤ 2
x + 2y ≥ 1
2. 3x – 2y ≤ –1
x + 4y ≥ –12
3. y ≤ 1
x>2
4. y ≥ 2 – 3
y < 2x
5. y < 3 + 2
y < –2x + 1
6. y ≥ – 4 + 1
y < 3x – 1
7. x + y ≥ 4
2x – y > 2
8. x + 3y < 3
x – 2y ≥ 4
9. x – 2y > 6
x + 4y < –4
Chapter 3
12
Glencoe Algebra 2
NAME _____________________________________________ DATE ____________________________ PERIOD _____________
3-2 Study Guide and Intervention
(continued)
Solving Systems of Inequalities by Graphing
Find Vertices of an Enclosed Region Sometimes the graph of a system of inequalities produces an enclosed region in
the form of a polygon. You can find the vertices of the region by a combination of the methods used earlier in this
chapter: graphing, substitution, and/or elimination.
Example: Find the coordinates of the vertices of the triangle formed by 5x + 4y < 20, y < 2x + 3, and x 3y < 4.
Graph each inequality. The intersections of the boundary lines are the
vertices of a triangle. The vertex (4, 0) can be determined from the graph.
To find the coordinates of the second and third vertices, solve the two
systems of equations
𝑦
2𝑥
3
𝑦
2𝑥
3
and
𝑥
3𝑦
4
5𝑥
4𝑦
20
For the first system of equations, rewrite the first equation
in standard form as 2x – y = –3. Then multiply that
equation by 4 and add to the second equation.
2x – y = –3
Multiply by 4.
8x – 4y = –12
5x + 4y = 20
(+) 5x + 4y = 20
13x
= 8
x
Then substitute x =
solve for y.
2
3
=
For the second system of equations, use substitution.
Substitute 2x + 3 for y in the second equation to get
x – 3(2x + 3) = 4
x – 6x – 9 = 4
–5x = 13
x=–
3
Then substitute x = –
in one of the original equations and
3
y=–
y=–
– y = –3
y=
3
5
y=2
– y = –3
3
3
5
2
5
3
5
in the first equation to solve for y.
+3
+3
5
The coordinates of the third vertex are
55
3
The coordinates of the second vertex are
3
,4
3
3
3
25, 25 .
.
Thus, the coordinates of the three vertices are (4, 0),
3
,4
3
3
and
3
25, 25 .
Exercises
Find the coordinates of the vertices of the triangle formed by each system of inequalities.
1. y ≤ –3x + 7
2. x > –3
3. y < – 2x + 3
y < 2x
y < – 3x + 3
y > 2x + 1
y > –2
y>x–1
y < 3x + 10
Chapter 3
13
Glencoe Algebra 2