Sure, let's solve each of these problems step by step.
### Problem a:
Statement: If 12 is added to a certain number, the answer is 35.
Number Sentence: Let the certain number be [tex]\( x \)[/tex]. The equation is:
[tex]\[ x + 12 = 35 \][/tex]
Solution:
To find [tex]\( x \)[/tex], we need to isolate the variable by subtracting 12 from both sides of the equation:
[tex]\[ x = 35 - 12 \][/tex]
[tex]\[ x = 23 \][/tex]
Check:
[tex]\[ 23 + 12 = 35 \][/tex]
The solution is correct.
### Problem b:
Statement: If 45 is subtracted from a certain number, the answer is 52.
Number Sentence: Let the certain number be [tex]\( y \)[/tex]. The equation is:
[tex]\[ y - 45 = 52 \][/tex]
Solution:
To find [tex]\( y \)[/tex], we need to isolate the variable by adding 45 to both sides of the equation:
[tex]\[ y = 52 + 45 \][/tex]
[tex]\[ y = 97 \][/tex]
Check:
[tex]\[ 97 - 45 = 52 \][/tex]
The solution is correct.
### Problem c:
Statement: A certain number is divided by 12. The answer is 7.
Number Sentence: Let the certain number be [tex]\( z \)[/tex]. The equation is:
[tex]\[ \frac{z}{12} = 7 \][/tex]
Solution:
To find [tex]\( z \)[/tex], we need to isolate the variable by multiplying both sides of the equation by 12:
[tex]\[ z = 7 \times 12 \][/tex]
[tex]\[ z = 84 \][/tex]
Check:
[tex]\[ \frac{84}{12} = 7 \][/tex]
The solution is correct.
### Problem d:
Statement: When 6 is multiplied by a certain number, the answer is 54.
Number Sentence: Let the certain number be [tex]\( w \)[/tex]. The equation is:
[tex]\[ 6 \times w = 54 \][/tex]
Solution:
To find [tex]\( w \)[/tex], we need to isolate the variable by dividing both sides of the equation by 6:
[tex]\[ w = \frac{54}{6} \][/tex]
[tex]\[ w = 9 \][/tex]
Check:
[tex]\[ 6 \times 9 = 54 \][/tex]
The solution is correct.
### Problem e:
Statement: 3 is multiplied by a certain number. 4 is added to this product. The answer is 19.
Number Sentence: Let the certain number be [tex]\( v \)[/tex]. The equation is:
[tex]\[ 3 \times v + 4 = 19 \][/tex]
Solution:
First, isolate the term with the variable by subtracting 4 from both sides:
[tex]\[ 3 \times v = 19 - 4 \][/tex]
[tex]\[ 3 \times v = 15 \][/tex]
Next, solve for [tex]\( v \)[/tex] by dividing both sides by 3:
[tex]\[ v = \frac{15}{3} \][/tex]
[tex]\[ v = 5 \][/tex]
Check:
[tex]\[ 3 \times 5 + 4 = 19 \][/tex]
The solution is correct.
### Summary of Solutions:
a) [tex]\( x = 23 \)[/tex]
b) [tex]\( y = 97 \)[/tex]
c) [tex]\( z = 84 \)[/tex]
d) [tex]\( w = 9 \)[/tex]
e) [tex]\( v = 5 \)[/tex]
Each solution has been verified and is correct.
