Download system of equations

10-6
10-6Systems
SystemsofofEquations
Equations
HOMEWORK & Learning Goal
Lesson Presentation
AIMS Prep
Pre-Algebra
Pre-Algebra
PA HOMEWORK Answers
Page 521
#1-11 ALL
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10-6 Systems of Equations
Don’t forget your proper heading! Trade & Grade!
10-5 Lesson Quiz: Part 1
Solve for the indicated variable.
1. P = R – C for C.
C=R-P
2. P = 2l+ 2w for l. P – 2w= l
2
3V = h
3. V = 1 Ah for h.
3
A
4. R = C – S for S. C – Rt = S
t
Pre-Algebra
10-6 Systems of Equations
Lesson Quiz: Part 2
5. Solve for y and graph 2x + 7y = 14.
y = – 2x + 2
7
4
y
2
–
4
Pre-Algebra
–
2
2
–
2
–
4
4
10-6 Systems of Equations
Pre-Algebra HOMEWORK
Page 526
#17-32
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Pre-Algebra
10-6 Systems of Equations
Our Learning Goal
Students will be able to solve
multi-step equations with
multiple variables, solve
inequalities and graph the
solutions on a number line.
Pre-Algebra
10-6 Systems of Equations
Our Learning Goal Assignments
• Learn to solve two-step equations.
• Learn to solve multistep equations.
• Learn to solve equations with variables on both
sides of the equal sign.
• Learn to solve two-step inequalities and graph
the solutions of an inequality on a number line.
• Learn to solve an equation for a variable.
• Learn to solve systems of equations.
Pre-Algebra
10-6 Systems of Equations
Today’s Learning Goal Assignment
Learn to solve
systems of
equations.
Pre-Algebra
10-6 Systems of Equations
Vocabulary
system of equations
solution of a system of equations
Pre-Algebra
10-6 Systems of Equations
A system of equations is a set of two or
more equations that contain two or more
variables. A solution of a system of
equations is a set of values that are solutions
of all of the equations. If the system has two
variables, the solutions can be written as
ordered pairs.
Pre-Algebra
10-6 Systems of Equations
Additional Example 1A: Identifying Solutions of a
System of Equations
Determine if the ordered pair is a solution of
the system of equations below.
5x + y = 7
x – 3y = 11
A. (1, 2)
5x + y = 7
?
5(1) + 2 = 7
7 = 7
x – 3y = 11
?
1 – 3(2) = 11
–5  11 
Substitute
for x and y.
The ordered pair (1, 2) is not a solution of
the system of equations.
Pre-Algebra
10-6 Systems of Equations
Try This: Example 1A
Determine if each ordered pair is a solution of
the system of equations below.
4x + y = 8
x – 4y = 12
A. (1, 2)
4x + y = 8
?
4(1) + 2 = 8
68 
x – 4y = 12
?
1 – 4(2) = 12
–7  12 
Substitute
for x and y.
The ordered pair (1, 2) is not a solution of
the system of equations.
Pre-Algebra
10-6 Systems of Equations
Additional Example 1B: Identifying Solutions of a
System of Equations
Determine if the ordered pair is a solution of
the system of equations below.
5x + y = 7
x – 3y = 11
B. (2, –3)
5x + y = 7
?
5(2) + –3 = 7
7 = 7
x – 3y = 11
?
2 – 3(–3) = 11
11 = 11 
Substitute
for x and y.
The ordered pair (2, –3) is a solution of the
system of equations.
Pre-Algebra
10-6 Systems of Equations
Try This: Example 1B
Determine if each ordered pair is a solution of
the system of equations below.
4x + y = 8
x – 4y = 12
B. (2, –3)
4x + y = 8
?
4(2) + –3 = 8
58 
x – 4y = 12
?
2 – 4(–3) = 12
14  12 
Substitute
for x and y.
The ordered pair (2, –3) is not a solution of
the system of equations.
Pre-Algebra
10-6 Systems of Equations
Additional Example 1C: Identifying Solutions of a
System of Equations
Determine if the ordered pair is a solution of
the system of equations below.
5x + y = 7
x – 3y = 11
C. (20, 3)
5x + y = 7
?
5(20) + (3) = 7
103  7 
x – 3y = 11
?
20 – 3(3) = 11
11 = 11 
Substitute
for x and y.
The ordered pair (20, 3) is not a solution of
the system of equations.
Pre-Algebra
10-6 Systems of Equations
Try This: Example 1C
Determine if each ordered pair is a solution of
the system of equations below.
4x + y = 8
x – 4y = 12
C. (1, 4)
4x + y = 8
x – 4y = 12
?
?
Substitute
4(1) + 4 = 8
1 – 4(4) = 12
for x and y.
–15  12 
8 = 8
The ordered pair (1, 4) is not a solution of
the system of equations.
Pre-Algebra
10-6 Systems of Equations
Helpful Hint
When solving systems of
equations, remember to
find values for all of the
variables.
Pre-Algebra
10-6 Systems of Equations
Additional Example 2: Solving Systems of Equations
Solve the system of equations.
y=y
y=x–4
y=x–4
y = 2x – 9
y = 2x – 9
x – 4 = 2x – 9
Solve the equation to find x.
x – 4 = 2x – 9
–x
–x
Subtract x from both sides.
–4 = x – 9
+9
+9
5=x
Pre-Algebra
Add 9 to both sides.
10-6 Systems of Equations
Additional Example 2 Continued
To find y, substitute 5 for x in one of the original
equations.
y=x–4=5–4=1
The solution is (5, 1).
Check: Substitute 5 for x and 1 for y in each equation.
y=x–4
1=5–4
y = 2x – 9
?
1 = 2(5) – 9
1=1 
1 = 1
?
Pre-Algebra
10-6 Systems of Equations
Try This: Example 2
Solve the system of equations.
y=y
y=x–5
y=x–5
y = 2x – 8
y = 2x – 8
x – 5 = 2x – 8
Solve the equation to find x.
x – 5 = 2x – 8
–x
–x
Subtract x from both sides.
–5 = x – 8
+8
+8
3=x
Pre-Algebra
Add 8 to both sides.
10-6 Systems of Equations
Try This: Example 2 Continued
To find y, substitute 3 for x in one of the original
equations.
y = x – 5 = 3 – 5 = –2
The solution is (3, –2).
Check: Substitute 3 for x and –2 for y in each
equation.
y=x–5
?
–2 = 3 – 5
–2 = –2 
Pre-Algebra
y = 2x – 8
?
–2 = 2(3) – 8
–2 = –2 
10-6 Systems of Equations
To solve a general
system of two equations
with two variables, you
can solve both equations
for x or both for y.
Pre-Algebra
10-6 Systems of Equations
Additional Example 3A: Solving Systems of
Equations
Solve the system of equations.
A. x + 2y = 8
x + 2y = 8
–2y
–2y
x
= 8 – 2y
Solve both
equations
for x.
8 – 2y = 13 + 3y
+ 2y
+ 2y
8
Pre-Algebra
= 13 + 5y
x – 3y = 13
x – 3y = 13
+ 3y
+ 3y
x
= 13 + 3y
Add 2y to
both sides.
10-6 Systems of Equations
Additional Example 3A Continued
8
–13
–5
= 13 + 5y
–13
=
5y
–5 = 5y
5
5
–1 = y
Subtract 13 from
both sides.
Divide both sides
by 5.
x = 8 – 2y
= 8 – 2(–1)
Substitute –1 for y.
= 8 + 2 = 10
The solution is (10, –1).
Pre-Algebra
10-6 Systems of Equations
Try This: Example 3A
Solve the system of equations.
A. x + y = 5
x+y=5
–x
–x
Solve both
equations
for y.
y=5–x
5 – x = –1 – 3x
+x
+ x
5
Pre-Algebra
= –1 – 2x
3x + y = –1
3x + y = –1
– 3x
– 3x
y = –1 – 3x
Add x to
both sides.
10-6 Systems of Equations
Try This: Example 3A Continued
5
+1
6
= –1 – 2x
+1
=
–2x
–3 = x
Add 1 to both sides.
Divide both sides
by –2.
y=5–x
= 5 – (–3)
Substitute –3 for x.
=5+3=8
The solution is (–3, 8).
Pre-Algebra
10-6 Systems of Equations
Helpful Hint
You can choose either
variable to solve for. It is
usually easiest to solve for a
variable that has a coefficient
of 1.
Pre-Algebra
10-6 Systems of Equations
Additional Example 3B: Solving Systems of
Equations
Solve the system of equations.
B. 3x – 3y = -3
3x – 3y = –3
–3x
–3x
–3y = –3 – 3x
Solve both 2x + y = -5
equations 2x + y = –5
–2x
–2x
for y.
y = –5 – 2x
–3y = –3 – 3x
–3
–3 –3
y=1+x
1 + x = –5 – 2x
Pre-Algebra
10-6 Systems of Equations
Additional Example 3B Continued
1 + x = –5 – 2x
+ 2x
+ 2x
1 + 3x = –5
–1
–1
3x = –6
3x = –6
3
3
x = –2
Add 2x to both
sides.
Subtract 1 from
both sides.
Divide both sides
by 3.
y=1+x
Substitute –2 for x.
= 1 + –2 = –1
The solution is (–2, –1).
Pre-Algebra
10-6 Systems of Equations
Try This: Example 3B
Solve the system
B. x + y = –2
x + y = –2
–x
–x
y = –2 – x
of equations.
Solve both
equations
for y.
–2 – x = 2 + 3x
Pre-Algebra
–3x + y = 2
–3x + y = 2
+ 3x
+ 3x
y = 2 + 3x
10-6 Systems of Equations
Try This: Example 3B Continued
–2 – x = 2 + 3x
+x
+x
–2
–2
= 2 + 4x
–2
–4
=
4x
–1 = x
y = 2 + 3x
= 2 + 3(–1) = –1
The solution is (–1, –1).
Pre-Algebra
Add x to both
sides.
Subtract 2 from
both sides.
Divide both sides
by 4.
Substitute –1
for x.
10-6 Systems of Equations
Don’t forget your proper heading! Trade & Grade!
10-6 Lesson Quiz
1. Determine if the ordered pair (2, 4) is a
solution of the system. y = 2x; y = –4x + 12
yes
Solve each system of equations.
1
(2 , 2)
3. 6x – y = –15; 2x + 3y = 5 (–2,3)
2. y = 2x + 1; y = 4x
4. Two numbers have a some of 23 and a
difference of 7. Find the two numbers.
15 and 8
Pre-Algebra