Reviewing Equations

1.

B. \(8=18−y\)

C. \(60=6y\)

For each equation, \(y = 10\) is a solution if it makes the equation true.

To test the each equation, we can follow these steps:

1. Substitute \(y= 10\) into the equation.
2. Simplify.
3. Check if both sides of the equation have the same value.

For example, here's how we could test the first equation:

\(\begin{array}{r}

y+11=22 \\

10+11 \stackrel{?}{=} 22 \\

21 \neq 22

\end{array}\)

No, \(y =10\) is not a solution.

\(y =10\) is a solution for the following equations:

• \(8=18−y\)
• \(60=6y\)

2.

C. \(f = 9\)

Let's substitute each \(f\) value into the equation to see if it is a solution.

Let's substitute \(f = 6\) and see if the equation is true.

\(\begin{array}{r}

3(f)+5=32 \\

3(6)+5 \stackrel{?}{=} 32 \\

18+5 \stackrel{?}{=} 32 \\

23 \neq 32

\end{array}\)

No, \(f = 6\) does not make a true statement.

Let's substitute \(f = 8\) and see if the equation is true.

\(\begin{array}{r}

3(f)+5=32 \\

3(8)+5 \stackrel{?}{=} 32 \\

24+5 \stackrel{?}{=} 32 \\

29 \neq 32

\end{array}\)

No, \(f = 8\) does not make a true statement.

Let's substitute \(f = 9\) and see if the equation is true.

\(\begin{array}{r}

3(f)+5=32 \\

3(9)+5 \stackrel{?}{=} 32 \\

27+5 \stackrel{?}{=} 32 \\

32 \stackrel{\checkmark}{=} 32

\end{array}\)

Yes, \(f = 9\) does make a true statement.

Let's substitute \(f = 12\) and see if the equation is true.

\(\begin{array}{r}

3(f)+5=32 \\

3(12)+5 \stackrel{?}{=} 32 \\

36+5 \stackrel{?}{=} 32 \\

41 \neq 32

\end{array}\)

No, \(f = 12\) does not make a true statement.

The \(f\) value that makes \(3f+5=32\) a true statement is \(f = 9 \).

3.

A. \(13=9+m\)

B. \(7−m=3\)

E. \(20÷m=5\)

For each equation, \(m = 4\) is a solution if it makes the equation true.

To test the each equation, we can follow these steps:

1. Substitute \( m = 4\) into the equation
2. Simplify.
3. Check if both sides of the equation have the same value.

For example, here's how we could test the first equation:

\(\begin{aligned}

&13=9+m \\

&13 \stackrel{?}{=} 9+4 \\

&13 \stackrel{\checkmark}{=} 13

\end{aligned}\)

Yes, \( m = 4\) is a solution.

\(m = 4\) is a solution for the following equations:

• \(13=9+m\)
• \(7−m=3\)
• \(20÷m=5\)

4.

B. \(h = 2\)

Let's substitute each \(h\) value into the equation to see if it is a solution.

Let's substitute \(h = 1\) and see if the equation is true.

\(\begin{aligned}

\frac{8+h}{10} &=1 \\

\frac{8+1}{10} & \stackrel{?}{=} 1 \\

\frac{9}{10} & \stackrel{?}{=} 1 \\

\frac{9}{10} & \neq 1

\end{aligned}\)

No, \(h = 1\) does not make a true statement.

Let's substitute \(h = 2\) and see if the equation is true.

\(\begin{array}{r}

\frac{8+h}{10}=1 \\

\frac{8+2}{10} \stackrel{?}{=} 1 \\

\frac{10}{10} \stackrel{?}{=} 1 \\

1 \stackrel{\checkmark}{=} 1

\end{array}\)

Yes, \(h = 2\) does make a true statement.

Let's substitute \(h = 3\) and see if the equation is true.

\(\begin{array}{r}

\frac{8+h}{10}=1 \\

\frac{8+3}{10} \stackrel{?}{=} 1 \\

\frac{11}{10} \stackrel{?}{=} 1 \\

\frac{11}{10} \neq 1

\end{array}\)

No, \(h = 3\) does not make a true statement.

Let's substitute \(h = 4\) and see if the equation is true.

\(\begin{array}{r}

\frac{8+h}{10}=1 \\

\frac{8+4}{10} \stackrel{?}{=} 1 \\

\frac{12}{10} \stackrel{?}{=} 1 \\

\frac{6}{5} \neq 1

\end{array}\)

No, \(h = 4\) does not make a true statement.

The \(h\)-value that makes \(\frac{8+h}{10}=1\) a true statement is \(h =2 \).

5.

C. \(77=7b\)

D. \(9=b−2\)

For each equation, \(b = 11\) is a solution if it makes the equation true.

To test the each equation, we can follow these steps:

1. Substitute \(b = 11\) into the equation.
2. Simplify.
3. Check if both sides of the equation have the same value.

For example, here's how we could test the first equation:

\(\begin{array}{r}

2 b=211 \\

2 \times 11 \stackrel{?}{=} 211 \\

22 \neq 211

\end{array}\)

No, \(b = 11\) is not a solution.

\(b = 11\) is a solution for the following equations:

• \(77=7b\)
• \(9=b−2\)

6.

A. \(g = 11\)

Let's substitute each \(g\)-value into the equation to see if it is a solution.

Let's substitute \(g = 11\) and see if the equation is true.

\(\begin{aligned}

&26=7(g-9)+12 \\

&26 \stackrel{?}{=} 7(11-9)+12 \\

&26 \stackrel{?}{=} 7(2)+12 \\

&26 \stackrel{?}{=} 14+12 \\

&26 \stackrel{\checkmark}{=} 26

\end{aligned}\)

Yes, \(g = 11\) does make a true statement.

Let's substitute \(g = 12\) and see if the equation is true.

\(\begin{aligned}

&26=7(g-9)+12 \\

&26 \stackrel{?}{=} 7(12-9)+12 \\

&26 \stackrel{?}{=} 7(3)+12 \\

&26 \stackrel{?}{=} 21+12 \\

&26 \neq 33

\end{aligned}\)

No, \(g = 12\) does not make a true statement.

Let's substitute \(g = 13\) and see if the equation is true.

\(\begin{aligned}

&26=7(g-9)+12 \\

&26 \stackrel{?}{=} 7(13-9)+12 \\

&26 \stackrel{?}{=} 7(4)+12 \\

&26 \stackrel{?}{=} 28+12 \\

&26 \neq 40

\end{aligned}\)

No, \(g = 13\) does not make a true statement.

Let's substitute \(g = 14\) and see if the equation is true.

\(\begin{aligned}

&26=7(g-9)+12 \\

&26 \stackrel{?}{=} 7(14-9)+12 \\

&26 \stackrel{?}{=} 7(5)+12 \\

&26 \stackrel{?}{=} 35+12 \\

&26 \neq 47

\end{aligned}\)

No, \(g = 14\) does not make a true statement.

The \(g\)- value that makes \(26=7(g-9)+12 \) a true statement is \(g = 11\).

7.

D. \(\frac{c}{3}= 3\)

E. \(36=4c\)

For each equation, \(c = 9\) is a solution if it makes the equation true.

To test the each equation, we can follow these steps:

1. Substitute \(c = 9\) into the equation.
2. Simplify.
3. Check if both sides of the equation have the same value.

For example, here's how we could test the first equation:

\(\begin{gathered}

4-c=5 \\

4-9 \stackrel{?}{=} 5 \\

-5 \neq 5

\end{gathered}\)

No, \(c = 9 \) is not a solution.

\( c = 9\) is a solution for the following equations:

• \(\frac {c}{3} = 3\)
• \(36=4c\)