Reviewing Equations
Testing solutions to equations - Questions
Answers
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:
- Substitute \(y= 10\) into the equation.
- Simplify.
- 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:
- Substitute \( m = 4\) into the equation
- Simplify.
- 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:
- Substitute \(b = 11\) into the equation.
- Simplify.
- 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:
- Substitute \(c = 9\) into the equation.
- Simplify.
- 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:
- \(\frac {c}{3} = 3\)
- \(36=4c\)