2.2 Solve equations using the division and multiplication properties (Page 2/4)

Elementary algebra Page 2 / 4

The multiplication property of equality

For any numbers a , b , and c ,

\begin{matrix} If & a & = & b, \\ then & a c & = & b c \end{matrix}

If you multiply both sides of an equation by the same number, you still have equality.

Solve: $\frac{y}{−7} = −14 .$

Solution

Here $y$ is divided by $−7$ . We must multiply by $−7$ to isolate $y$ .


Multiply both sides by $−7$ .
Multiply.
Simplify.
Check: $\frac{y}{−7} = −14$
Substitute $y = 98$ .
Divide.

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Solve: $\frac{a}{−7} = −42 .$

$a = 294$

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Solve: $\frac{b}{−6} = −24 .$

$b = 144$

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Solve: $− n = 9 .$

Solution


Remember $− n$ is equivalent to $−1 n$ .
Divide both sides by $−1$ .
Divide.
Notice that there are two other ways to solve $− n = 9$ . We can also solve this equation by multiplying both sides by $−1$ and also by taking the opposite of both sides.
Check:
Substitute $n = −9$ .
Simplify.

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Solve: $− k = 8 .$

$k = −8$

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Solve: $− g = 3 .$

$g = −3$

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Solve: $\frac{3}{4} x = 12 .$

Solution

Since the product of a number and its reciprocal is 1, our strategy will be to isolate $x$ by multiplying by the reciprocal of $\frac{3}{4}$ .


Multiply by the reciprocal of $\frac{3}{4}$ .
Reciprocals multiply to 1.
Multiply.
Notice that we could have divided both sides of the equation $\frac{3}{4} x = 12$ by $\frac{3}{4}$ to isolate $x$ . While this would work, most people would find multiplying by the reciprocal easier.
Check:
Substitute $x = 16$ .

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Solve: $\frac{2}{5} n = 14 .$

$n = 35$

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Solve: $\frac{5}{6} y = 15 .$

$y = 18$

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In the next example, all the variable terms are on the right side of the equation. As always, our goal in solving the equation is to isolate the variable.

Solve: $\frac{8}{15} = - \frac{4}{5} x .$

Solution


Multiply by the reciprocal of $- \frac{4}{5}$ .
Reciprocals multiply to 1.
Multiply.
Check:
Let $x = - \frac{2}{3}$ .

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Solve: $\frac{9}{25} = - \frac{4}{5} z .$

$z = - \frac{9}{5}$

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Solve: $\frac{5}{6} = - \frac{8}{3} r .$

$r = - \frac{5}{16}$

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Solve equations that require simplification

Many equations start out more complicated than the ones we have been working with.

With these more complicated equations the first step is to simplify both sides of the equation as much as possible. This usually involves combining like terms or using the distributive property.

Solve: $14 - 23 = 12 y - 4 y - 5 y .$

Solution

Begin by simplifying each side of the equation.


Simplify each side.
Divide both sides by $3$ to isolate $y$ .
Divide.
Check:
Substitute $y = −3$ .

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Solve: $18 - 27 = 15 c - 9 c - 3 c .$

$c = −3$

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Solve: $18 - 22 = 12 x - x - 4 x .$

$x = - \frac{4}{7}$

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Solve: $−4 (a - 3) - 7 = 25 .$

Solution

Here we will simplify each side of the equation by using the distributive property first.


Distribute.
Simplify.
Simplify.
Divide both sides by $−4$ to isolate $a$ .
Divide.
Check:
Substitute $a = −5$ .

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Solve: $−4 (q - 2) - 8 = 24 .$

$q = −6$

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Solve: $−6 (r - 2) - 12 = 30 .$

$r = −5$

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Now we have covered all four properties of equality—subtraction, addition, division, and multiplication. We’ll list them all together here for easy reference.

Properties of equality

\begin{matrix} Subtraction Property of Equality & Addition Property of Equality \\ For any real numbers a, b, and c, & For any real numbers a, b, and c, \\ \begin{matrix} if & a & = & b, \\ then & a - c & = & b - c . \end{matrix} & \begin{matrix} if & a & = & b, \\ then & a + c & = & b + c . \end{matrix} \\ Division Property of Equality & Multiplication Property of Equality \\ For any numbers a, b, and c, and c \neq 0, & For any numbers a, b, and c, \\ \begin{matrix} if & a & = & b, \\ then & \frac{a}{c} & = & \frac{b}{c} . \end{matrix} & \begin{matrix} if & a & = & b, \\ then & a c & = & b c . \end{matrix} \end{matrix}

When you add, subtract, multiply, or divide the same quantity from both sides of an equation, you still have equality.

Translate to an equation and solve

In the next few examples, we will translate sentences into equations and then solve the equations. You might want to review the translation table in the previous chapter.

Questions & Answers

how did you get 1640

Noor Reply

If auger is pair are the roots of equation x2+5x-3=0

Peter Reply

Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.

MATTHEW Reply

420

Sharon

from theory: distance [miles] = speed [mph] × time [hours] info #1 speed_Dennis × 1.5 = speed_Wayne × 2 => speed_Wayne = 0.75 × speed_Dennis (i) info #2 speed_Dennis = speed_Wayne + 7 [mph] (ii) use (i) in (ii) => [...] speed_Dennis = 28 mph speed_Wayne = 21 mph

George

Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5. Substituting the first equation into the second: W * 2 = (W + 7) * 1.5 W * 2 = W * 1.5 + 7 * 1.5 0.5 * W = 7 * 1.5 W = 7 * 3 or 21 W is 21 D = W + 7 D = 21 + 7 D = 28

Salma

Devon is 32 32 years older than his son, Milan. The sum of both their ages is 54 54. Using the variables d d and m m to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.

Aaron Reply

find product (-6m+6) ( 3m²+4m-3)

SIMRAN Reply

-42m²+60m-18

Salma

what is the solution

bill

how did you arrive at this answer?

bill

-24m+3+3mÁ^2

Susan

i really want to learn

Amira

I only got 42 the rest i don't know how to solve it. Please i need help from anyone to help me improve my solving mathematics please

Amira

Hw did u arrive to this answer.

Aphelele

Bajemah

-6m(3mA²+4m-3)+6(3mA²+4m-3) =-18m²A²-24m²+18m+18mA²+24m-18 Rearrange like items -18m²A²-24m²+42m+18A²-18

Salma

complete the table of valuesfor each given equatio then graph. 1.x+2y=3

Jovelyn Reply

x=3-2y

Salma

y=x+3/2

Salma

Enock

given that (7x-5):(2+4x)=8:7find the value of x

Nandala

3x-12y=18

Kelvin

please why isn't that the 0is in ten thousand place

Grace Reply

please why is it that the 0is in the place of ten thousand

Grace

Send the example to me here and let me see

Stephen

A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg

Marry Reply

how far

Abubakar

cool u

Enock

state in which quadrant or on which axis each of the following angles given measure. in standard position would lie 89°

Abegail Reply

hello

BenJay

Method

I am eliacin, I need your help in maths

Rood

how can I help

Sir

hmm can we speak here?

Amoon

however, may I ask you some questions about Algarba?

Amoon

Enock

what the last part of the problem mean?

Roger

The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.

cameron Reply

Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.

mahnoor Reply

I'm guessing, but it's somewhere around $4335.00 I think

Lewis

12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67. Check: Sales = 3542 Commission 12%=425.04 Pay = 500 + 425.04 = 925.04. 925.04 > 925.00

Munster

difference between rational and irrational numbers

Arundhati Reply

When traveling to Great Britain, Bethany exchanged $602 US dollars into £515 British pounds. How many pounds did she receive for each US dollar?

Jakoiya Reply

how to reduced echelon form

Solomon Reply

Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?

Zack Reply

d=r×t the equation would be 8/r+24/r+4=3 worked out

Sheirtina

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Source: OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2

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