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EXERCISE 7.1

Write an algebraic equation for each problem, then solve it.

a) The sum of 5 times a number and 3 is 43.
b) The difference between 64 and a number is equal to 48.
c) The product of a number and 12 is 108.
d) 54 subtracted from double a certain number is 12.
e) The difference between five times a number and 12 is 28.
f) The quotient of 24 and a number is 8. What is the number?

Find the value of the variable that makes these number sentences true:

a) [tex]\(15 + a = 28\)[/tex]
b) [tex]\(b - 35 = 27\)[/tex]
c) [tex]\(d \div 8 = 12\)[/tex]
d) [tex]\(3e + 14 = 26\)[/tex]
e) [tex]\(3c = 69\)[/tex]
f) [tex]\(2f - 10 = ?\)[/tex]



Answer :

GinnyAnswer
Let's solve each algebraic equation step-by-step:

### 1. The sum of 5 times a number and 3 is 43.

Form the equation:
[tex]\[ 5a + 3 = 43 \][/tex]

Subtract 3 from both sides:
[tex]\[ 5a = 40 \][/tex]

Divide both sides by 5:
[tex]\[ a = 8 \][/tex]

### 2. The difference between 64 and a number is equal to 48.

Form the equation:
[tex]\[ 64 - b = 48 \][/tex]

Subtract 64 from both sides and then simplify:
[tex]\[ -b = 48 - 64 \][/tex]
[tex]\[ -b = -16 \][/tex]

Multiply both sides by -1:
[tex]\[ b = 16 \][/tex]

### 3. The product of a number and 12 is 108.

Form the equation:
[tex]\[ 12c = 108 \][/tex]

Divide both sides by 12:
[tex]\[ c = 9 \][/tex]

### 4. 54 subtracted from double a certain number is 12.

Form the equation:
[tex]\[ 2d - 54 = 12 \][/tex]

Add 54 to both sides:
[tex]\[ 2d = 66 \][/tex]

Divide both sides by 2:
[tex]\[ d = 33 \][/tex]

### 5. The difference between five times a number and 12 is 28.

Form the equation:
[tex]\[ 5e - 12 = 28 \][/tex]

Add 12 to both sides:
[tex]\[ 5e = 40 \][/tex]

Divide both sides by 5:
[tex]\[ e = 8 \][/tex]

### 6. The quotient of 24 and a number is 8.

Form the equation:
[tex]\[ \frac{24}{f} = 8 \][/tex]

Multiply both sides by [tex]\( f \)[/tex] and then divide by 8:
[tex]\[ 24 = 8f \][/tex]
[tex]\[ f = 3 \][/tex]

### Finding the value of the variable in the following number sentences:

#### 1. [tex]\( 15 + a = 28 \)[/tex]

Subtract 15 from both sides:
[tex]\[ a = 13 \][/tex]

#### 2. [tex]\( b - 35 = 27 \)[/tex]

Add 35 to both sides:
[tex]\[ b = 62 \][/tex]

#### 3. [tex]\( d \div 8 = 12 \)[/tex]

Multiply both sides by 8:
[tex]\[ d = 96 \][/tex]

#### 4. [tex]\( 3e + 14 = 26 \)[/tex]

Subtract 14 from both sides:
[tex]\[ 3e = 12 \][/tex]

Divide both sides by 3:
[tex]\[ e = 4 \][/tex]

#### 5. [tex]\( 3c = 69 \)[/tex]

Divide both sides by 3:
[tex]\[ c = 23 \][/tex]

#### 6. [tex]\( 2f - 10 = 0 \)[/tex]

Add 10 to both sides:
[tex]\[ 2f = 10 \][/tex]

Divide both sides by 2:
[tex]\[ f = 5 \][/tex]

### Summary of Solved Values

Here are the values for each variable based on the equations:

- [tex]\( a = 8 \)[/tex]
- [tex]\( b = 16 \)[/tex]
- [tex]\( c = 9 \)[/tex]
- [tex]\( d = 33 \)[/tex]
- [tex]\( e = 8 \)[/tex]
- [tex]\( f = 3 \)[/tex]

And the values for the variables in the given number sentences:

- [tex]\( a = 13 \)[/tex]
- [tex]\( b = 62 \)[/tex]
- [tex]\( d = 96 \)[/tex]
- [tex]\( e = 4 \)[/tex]
- [tex]\( c = 23 \)[/tex]
- [tex]\( f = 5 \)[/tex]

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