PDF 1.3 Solving Equations with Variables on Both Sides
1.3
Solving Equations with Variables
on Both Sides
How can you solve an equation that has
variables on both sides?
1
ACTIVITY: Perimeter and Area
Work with a partner.
Each figure has the unusual property that the value of its perimeter
(in feet) is equal to the value of its area (in square feet). Write an
equation for each figure.
Solve each equation for x.
Use the value of x to find the perimeter and the area of each figure.
Describe how you can check your solution.
b.
a.
3
x
x
c.
4
d.
x
5
2
18
x
e.
f.
1
4
2
2
3
COMMON
CORE
Solving Equations
In this lesson, you will
solve equations with
variables on both sides.
determine whether
equations have no solution
or infinitely many solutions.
Learning Standards
8.EE.7a
8.EE.7b
x1
x
g.
3x
x
1
3
2
1
x
18
Chapter 1
Equations
x
Lauries Notes
Introduction
Standards for Mathematical Practice
?
MP6 Attend to Precision: Mathematically proficient students try to
communicate precisely with others. In todays activities, students will work
with vocabulary related to measurement.
Motivate
What balances with the cylinder? Explain.
Common Core State Standards
8.EE.7a Give examples of linear
equations in one variable with one
solution, infinitely many solutions, or
no solutions. Show which of these
possibilities is the case by successively
transforming the given equation into
simpler forms, until an equivalent
equation of the form x = a, a = a,
or a = b results (where a and b are
different numbers).
8.EE.7b Solve linear equations with
rational number coefficients, including
equations whose solutions require
expanding expressions using the
distributive property and collecting
like terms.
Previous Learning
?
?
2 cubes; Remove one cube and one cylinder from each side. 2 cylinders
balance with 4 cubes, so 1 cylinder would balance with 2 cubes.
The balance problem is equivalent to x + 5 = 3x + 1, where x is a cylinder
and the whole numbers represent cubes. This is an example of an equation
with variables on both sides, the type students will solve today. Return to
this equation at the end of class.
MP4 Model with Mathematics: If students are familiar with algebra tiles,
you can model the problem using the tiles. The cylinder is replaced with an
x-tile, and the cubes are replaced with unit tiles.
Students should know common formulas
for perimeter, area, surface area,
and volume.
Technology
ogy for the
Teacher
Lesson Plans
Complete Materials List
Activity Notes
Activity 1
?
?
?
Discuss with students the general concept of what it means to measure
the attributes of a two-dimensional figure. In other words, what is the
difference between a rectangles perimeter and a rectangles area? What
type of units are used to measure each? linear units for perimeter and
square units for area
MP6: Be sure to make it clear that the directions are saying that perimeter
and area are not the same, but their values are equal. For example, a
square that measures 4 centimeters on each edge has a perimeter of
16 centimeters and an area of 16 square centimeters. The value (16) is the
same, but the units of measure are not.
Before students begin, ask a few review questions.
? How do you find the perimeter P and the area A of a rectangle with
length ? and width w ? P = 2? + 2w and A = ?w
How
do you find the perimeter and the area of a composite
?
figure? Listen for students understanding that perimeter is the sum
of all of the sides. The area is found in parts and then added together.
Have a few groups share their work at the board, particularly for part (d),
fractions, and part (g), algebraic expressions.
1.3 Record and Practice Journal
Essential Question How can you solve an equation that has variables
on both sides?
1
ACTIVITY: Perimeter and Area
Work with a partner.
?
Each figure has the unusual property that the value of its perimeter
(in feet) is equal to the value of its area (in square feet).Write an
equation for each figure.
?
Solve each equation for x.
?
Use the value of x to find the perimeter and the area of each figure.
?
Describe how you can check your solution.
a.
b.
3
x
x
4
2x + 6 = 3x
x=6
P = 18 ft
A = 18 ft2
c. x
2x + 8 = 4x
x=4
P = 16 ft
A = 16 ft2
d.
5
2
18
2x + 36 = 18x
1
x = 2
4
P = 40.5 ft
A = 40.5 ft2
x
5
2
2x + 5 = x
x = 10
P = 25 ft
A = 25 ft2
T-18
Lauries Notes
Differentiated Instruction
Auditory
Point out to students that skills used to
solve equations in this lesson are the
same skills they have used before. The
goal is to isolate the variable on one
side of the equation. Just as they used
the Addition Property of Equality to
remove a constant term from one side
of the equation, they will use the same
property to remove the variable term
from one side of the equation.
Activity 2
?
?
Activity 3
?
?
1.3 Record and Practice Journal
e.
f.
1
4
2
2
3
This activity is similar to Activity 1.
How do you find the surface area S and volume V of a rectangular prism
with length ?, width w, and height h? S = 2?w + 2?h + 2wh and
V = ?wh
Students may guess that part (a) is a cube, suggesting x = 6. Ask students
to verify their guesses.
The larger triangle is a scale drawing of the smaller triangle. How can
you find the missing dimensions? Sample answer: Write a proportion to
solve for the side labeled x in the smaller triangle, then use this value to
find the side labeled 2x in the larger triangle.
The above described method is a good preview of similar figures in the
next chapter.
Another Way: There are other approaches students may take in solving
this problem.
? One method is to solve the equation: 150% of the smaller triangles
perimeter is equal to the perimeter of the larger triangle.
x+1
150% of (18 + x) = 24 + 2x
1.5(18 + x) = 24 + 2x
x
2x + 8 = 3x + 2
x=6
P = 20 ft
A = 20 ft2
g.
x
1
2
2x + 16 = 2x + 4(x + 1)
x=3
P = 22 ft
A = 22 ft2
3x
?
3
1
x
2
x
6x + 10 = 9x + x + x
x=2
P = 22 ft
A = 22 ft2
?
ACTIVITY: Surface Area and Volume
This method reviews decimal multiplication.
Another method is to find the scale factor, and then use it to determine
the missing dimensions.
Neighbor Check: Have students work independently and then have their
neighbors check their work. Have students discuss any discrepancies.
Work with a partner.
?
Each solid on the next page has the unusual property that the value
of its surface area (in square inches) is equal to the value of its volume
(in cubic inches).Write an equation for each solid.
?
Solve each equation for x.
?
Use the value of x to find the surface area and the volume of each solid.
?
Describe how you can check your solution.
Closure
?
a.
b.
x
4
8
6
6
x
8x + 16x + 64 = 32x
x=8
S = 256 in.2; V = 256 in.3
12x + 72 + 12x = 36x
x=6
S = 216 in.2; V = 216 in.3
3
ACTIVITY: Puzzle
Work with a partner. The perimeter of the larger triangle is 150% of the
perimeter of the smaller triangle. Find the dimensions of each triangle.
10
x
15
9
8
x=6
6, 8, 10
2x
2x = 12
9, 12, 15
What Is Your Answer?
4. IN YOUR OWN WORDS How can you solve an equation that has variables
on both sides? How do you move a variable term from one side of the
equation to the other?
Add or subtract variable terms and constant terms to
collect the variable terms on one side and constant
terms on the other side.
5. Write an equation that has variables on both sides. Solve the equation.
See Additional Answers.
T-19
Describe how to solve x + 5 = 3x + 1. Sample answer: Subtract x from
both sides, subtract 1 from both sides, and then divide both sides by 2.
2
ACTIVITY: Surface Area and Volume
Work with a partner.
Math
Practice
Each solid has the unusual property that the value of its surface area
(in square inches) is equal to the value of its volume (in cubic inches).
Write an equation for each solid.
Solve each equation for x.
Use the value of x to find the surface area and the volume of each solid.
Describe how you can check your solution.
Use Operations
What properties
of operations do
you need to use in
order to find the
value of x?
a.
b.
x
4
8
6
x
6
3
ACTIVITY: Puzzle
Work with a partner. The perimeter of the larger triangle is 150% of the
perimeter of the smaller triangle. Find the dimensions of each triangle.
10
x
15
9
8
2x
4. IN YOUR OWN WORDS How can you solve an equation that has variables
on both sides? How do you move a variable term from one side of the
equation to the other?
5. Write an equation that has variables on both sides. Solve the equation.
Use what you learned about solving equations with variables on
both sides to complete Exercises 3 C 5 on page 23.
Section 1.3
Solving Equations with Variables on Both Sides
19
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