4.7 Solve equations with fractions (Page 3/5)

Prealgebra

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\frac{3}{4} x = 24

The coefficient of $x$ is $\frac{3}{4} .$ To solve for $x,$ we need its coefficient to be $1 .$ Since the product of a number and its reciprocal is $1,$ our strategy here will be to isolate $x$ by multiplying by the reciprocal of $\frac{3}{4} .$ We will do this in [link] .

Solve: $\frac{3}{4} x = 24 .$

Solution


Multiply both sides by the reciprocal of the coefficient.
Simplify.
Multiply.
Check:
Substitute $x = 32$ .
Rewrite $32$ as a fraction.
Multiply. The equation is true.

Notice that in the equation $\frac{3}{4} x = 24,$ we could have divided both sides by $\frac{3}{4}$ to get $x$ by itself. Dividing is the same as multiplying by the reciprocal, so we would get the same result. But most people agree that multiplying by the reciprocal is easier.

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

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

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Solve: $- \frac{3}{8} w = 72 .$

Solution

The coefficient is a negative fraction. Remember that a number and its reciprocal have the same sign, so the reciprocal of the coefficient must also be negative.


Multiply both sides by the reciprocal of $− \frac{3}{8}$ .
Simplify; reciprocals multiply to one.
Multiply.
Check:
Let $w = −192$ .
Multiply. It checks.

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Solve: $- \frac{4}{7} a = 52 .$

−91

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Solve: $- \frac{7}{9} w = 84 .$

−108

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Translate sentences to equations and solve

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

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

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

In the next few examples, we’ll translate sentences into equations and then solve the equations. It might be helpful to review the translation table in Multiply Whole Numbers .

Translate and solve: $n$ divided by $6$ is $−24 .$

Solution

Translate.
Multiply both sides by $6$ .
Simplify.
Check:	Is $−144$ divided by $6$ equal to $−24$ ?
Translate.
Simplify. It checks.

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Translate and solve: $n$ divided by $7$ is equal to $−21 .$

$\frac{n}{7} = −21; n = −147$

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Translate and solve: $n$ divided by $8$ is equal to $−56 .$

$\frac{n}{8} = −56; n = −448$

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Translate and solve: The quotient of $q$ and $−5$ is $70 .$

Solution

Translate.
Multiply both sides by $−5$ .
Simplify.
Check:	Is the quotient of $−350$ and $−5$ equal to $70$ ?
Translate.
Simplify. It checks.

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Translate and solve: The quotient of $q$ and $−8$ is $72 .$

$\frac{q}{−8} = 72; q = −576$

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Translate and solve: The quotient of $p$ and $−9$ is $81 .$

$\frac{p}{−9} = 81; p = −729$

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Translate and solve: Two-thirds of $f$ is $18 .$

Solution

Translate.
Multiply both sides by $\frac{3}{2}$ .
Simplify.
Check:	Is two-thirds of $27$ equal to $18$ ?
Translate.
Simplify. It checks.

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Translate and solve: Two-fifths of $f$ is $16 .$

$\frac{2}{5} f = 16; f = 40$

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Translate and solve: Three-fourths of $f$ is $21 .$

$\frac{3}{4} f = 21; f = 28$

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Translate and solve: The quotient of $m$ and $\frac{5}{6}$ is $\frac{3}{4} .$

Solution

	The quotient of $m$ and $\frac{5}{6}$ is $\frac{3}{4}$ .
Translate.	$\frac{m}{\frac{5}{6}} = \frac{3}{4}$
Multiply both sides by $\frac{5}{6}$ to isolate $m$ .	$\frac{5}{6} (\frac{m}{\frac{5}{6}}) = \frac{5}{6} (\frac{3}{4})$
Simplify.	$m = \frac{5 \cdot 3}{6 \cdot 4}$
Remove common factors and multiply.	$m = \frac{5}{8}$
Check:
Is the quotient of $\frac{5}{8}$ and $\frac{5}{6}$ equal to $\frac{3}{4}$ ?	$\frac{\frac{5}{8}}{\frac{5}{6}} \overset{?}{=} \frac{3}{4}$
Rewrite as division.	$\frac{5}{8} \div \frac{5}{6} \overset{?}{=} \frac{3}{4}$
Multiply the first fraction by the reciprocal of the second.	$\frac{5}{8} \cdot \frac{6}{5} \overset{?}{=} \frac{3}{4}$
Simplify.	$\frac{3}{4} = \frac{3}{4} ✓$

Our solution checks.

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Questions & Answers

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is the branch of biology that deals with the study of microorganisms.

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studies of microbes

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they make spores

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the significance of food webs for disease transmission

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food webs brings about an infection as an individual depends on number of diseased foods or carriers dully.

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Assimilatory nitrate reduction is a process that occurs in some microorganisms, such as bacteria and archaea, in which nitrate (NO3-) is reduced to nitrite (NO2-), and then further reduced to ammonia (NH3).

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This process is called assimilatory nitrate reduction because the nitrogen that is produced is incorporated in the cells of microorganisms where it can be used in the synthesis of amino acids and other nitrogen products

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Prevent foreign microbes to the host

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They are friends to host only when Host immune system is strong and become enemies when the host immune system is weakened . very bad relationship!

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Source: OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9

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