section 2.5 solving linear equations in one variable using the multiplication-division principle
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Section 2.5
Solving Linear Equations in One Variable Using the Multiplication-Division Principle
2.5 Lecture Guide: Solving Linear Equations in One Variable Using the Multiplication-Division Principle
Objective 1: Solve linear equations in one variable using the multiplication-division principle.
Verbally
If both sides of an equation are multiplied or divided by the same nonzero number, the result is an ____________ equation.
Algebraically
If a, b, and c are real numbers and , then is equivalent to and to .
Numerical Example
is equivalent
to ; and
is equivalent to .
0c a b
_____ac
_____a
c
52
x
2 2 52
x
3 12x3 12
3 3
x
Solve each equation.
1. 2. 8 72x 73x
Strategy for Solving Linear EquationsStep 1: __________ each side of the equation.(a) If the equation contains fractions, simplify by __________ both sides of the equation by the least common denominator (LCD) of all the fractions.(b) If the equation contains grouping symbols, simplify by using the distributive property to remove the grouping symbols and then combine like terms.
Step 2: Using the addition-subtraction principle of equality, isolate the variable terms on one side of the equation and the _________ terms on the other side.
Step 3: Using the multiplication-division principle of equality, solve the equation produced in Step 2.
The following examples require using the multiplication-division principle of equality. Some will also require using the addition-subtraction principle of equality. Solve each equation. Note that we can check our solutions of each equation.
3. 2 6x 4. 2 6x
The following examples require using the multiplication-division principle of equality. Some will also require using the addition-subtraction principle of equality. Solve each equation. Note that we can check our solutions of each equation.
5. 6.62
x 2 6x
The following examples require using the multiplication-division principle of equality. Some will also require using the addition-subtraction principle of equality. Solve each equation. Note that we can check our solutions of each equation.
7. 8.12a 45x
The following examples require using the multiplication-division principle of equality. Some will also require using the addition-subtraction principle of equality. Solve each equation. Note that we can check our solutions of each equation.
9. 10.7 5 23x 8 1 3 23t t
11. 2 3 1 2 22x x
The following examples require using the multiplication-division principle of equality. Some will also require using the addition-subtraction principle of equality. Solve each equation. Note that we can check our solutions of each equation.
12. 3 4 5 5 2 1x x
The following examples require using the multiplication-division principle of equality. Some will also require using the addition-subtraction principle of equality. Solve each equation. Note that we can check our solutions of each equation.
13.
The following examples require using the multiplication-division principle of equality. Some will also require using the addition-subtraction principle of equality. Solve each equation. Note that we can check our solutions of each equation.
5 3 2 1 4 2x x
14.
The following examples require using the multiplication-division principle of equality. Some will also require using the addition-subtraction principle of equality. Solve each equation. Note that we can check our solutions of each equation.
1 43 2x x
15.
The following examples require using the multiplication-division principle of equality. Some will also require using the addition-subtraction principle of equality. Solve each equation. Note that we can check our solutions of each equation.
1 73
5 12m m
16.
The following examples require using the multiplication-division principle of equality. Some will also require using the addition-subtraction principle of equality. Solve each equation. Note that we can check our solutions of each equation.
5 1 3 21
3 5x x
17.
The following examples require using the multiplication-division principle of equality. Some will also require using the addition-subtraction principle of equality. Solve each equation. Note that we can check our solutions of each equation.
)43()32(4)2(3)1(6 xxxx
18. Note the difference between simplifying expressions and solving equations:
(a) Simplify
(b) Solve
3 5 2 2 8 1x x
3 5 2 2 8 1x x
Translate each verbal statement into algebraic form.
20.
19. Twice the sum of a number and 3 is equal to eleven less than four times the number.
One-half the quantity of a number plus three is the same as five minus the number.
21. Write an algebraic equation for the following statement and then solve the equation.
Verbal Statement: Three times the quantity of a number plus four is two less than the number.
Algebraic Equation:
Solve this equation:
22. The perimeter of the rectangle shown equals 44 cm.
Find a.
a cm a cm
7 cma
7 cma
23. Solve the equation by letting 1Yequal the left side of the equation and 2Y equal theright side of the equation.
(a) Use your calculator to create a graph of 1Y and 2Yusing a viewing window of 5, 5, 1 by 5, 5, 1 . Use theIntersect feature to find the point where these two lines intersect. Draw a rough sketch below. The values in the table will help.
-5
-4
-3
-2
-1
0
1
2
3
4
5
-5 -4 -3 -2 -1 0 1 2 3 4 5
y
x
The point where the two lines intersect has an x-coordinate of ______.
4 2 5x x
(b) Create a table on your calculator with the table settings: TblStart = 0;
x Y1 Y2
0
1
2
3
4
5
6
Tbl 1 Complete the table below.
The x-value at which the two y-values are equal is ______.
(c) Solve the equation 4 2 5x x algebraically.
(d) Check your solution.