Question

solve the equation. check your solution.

1. ( a - 6 = 13 )
2. ( -3 = z - 8 )
3. ( -14 = k + 6 )
4. ( x + 4 = -14 )
5. ( c - 7.6 = -4 )
6. ( -10.1 = w + 5.3 )
7. ( \frac{1}{2} = q + \frac{2}{3} )
8. ( p - 3\frac{1}{6} = -2\frac{1}{2} )
9. ( g - 9 = -19 )
10. ( -9.3 = d - 3.4 )
11. ( 4.58 + y = 2.5 )
12. ( x - 5.2 = -18.73 )
13. ( q + \frac{5}{9} = \frac{1}{6} )
14. ( -2\frac{1}{4} = r - \frac{4}{5} )
15. ( w + 3\frac{3}{8} = 1\frac{5}{6} )
16. ( 4\frac{2}{5} + k = -3\frac{2}{11} )

Explanation:

Let's solve equation 5: \( a - 6 = 13 \)

Step1: Isolate the variable \( a \)

To isolate \( a \), we add 6 to both sides of the equation.
\( a - 6 + 6 = 13 + 6 \)

Step2: Simplify both sides

Simplifying the left side gives \( a \), and simplifying the right side: \( 13 + 6 = 19 \)
So, \( a = 19 \)

Check:

Substitute \( a = 19 \) back into the original equation: \( 19 - 6 = 13 \), which is true.

Now equation 6: \( -3 = z - 8 \)

Step1: Isolate \( z \)

Add 8 to both sides: \( -3 + 8 = z - 8 + 8 \)

Step2: Simplify

Left side: \( -3 + 8 = 5 \), right side: \( z \)
So, \( z = 5 \)

Check:

Substitute \( z = 5 \) into the original equation: \( -3 = 5 - 8 \), and \( 5 - 8 = -3 \), which is true.

Equation 7: \( -14 = k + 6 \)

Step1: Isolate \( k \)

Subtract 6 from both sides: \( -14 - 6 = k + 6 - 6 \)

Step2: Simplify

Left side: \( -14 - 6 = -20 \), right side: \( k \)
So, \( k = -20 \)

Check:

Substitute \( k = -20 \) into the original equation: \( -14 = -20 + 6 \), and \( -20 + 6 = -14 \), which is true.

Equation 8: \( x + 4 = -14 \)

Step1: Isolate \( x \)

Subtract 4 from both sides: \( x + 4 - 4 = -14 - 4 \)

Step2: Simplify

Left side: \( x \), right side: \( -14 - 4 = -18 \)
So, \( x = -18 \)

Check:

Substitute \( x = -18 \) into the original equation: \( -18 + 4 = -14 \), which is true.

Equation 9: \( c - 7.6 = -4 \)

Step1: Isolate \( c \)

Add 7.6 to both sides: \( c - 7.6 + 7.6 = -4 + 7.6 \)

Step2: Simplify

Left side: \( c \), right side: \( -4 + 7.6 = 3.6 \)
So, \( c = 3.6 \)

Check:

Substitute \( c = 3.6 \) into the original equation: \( 3.6 - 7.6 = -4 \), which is true.

Equation 10: \( -10.1 = w + 5.3 \)

Step1: Isolate \( w \)

Subtract 5.3 from both sides: \( -10.1 - 5.3 = w + 5.3 - 5.3 \)

Step2: Simplify

Left side: \( -10.1 - 5.3 = -15.4 \), right side: \( w \)
So, \( w = -15.4 \)

Check:

Substitute \( w = -15.4 \) into the original equation: \( -10.1 = -15.4 + 5.3 \), and \( -15.4 + 5.3 = -10.1 \), which is true.

Equation 11: \( \frac{1}{2} = q + \frac{2}{3} \)

Step1: Isolate \( q \)

Subtract \( \frac{2}{3} \) from both sides: \( \frac{1}{2} - \frac{2}{3} = q + \frac{2}{3} - \frac{2}{3} \)

Step2: Find a common denominator for the left side

Common denominator of 2 and 3 is 6. So, \( \frac{3}{6} - \frac{4}{6} = q \)

Step3: Simplify

\( \frac{3 - 4}{6} = q \), so \( -\frac{1}{6} = q \)

Check:

Substitute \( q = -\frac{1}{6} \) into the original equation: \( \frac{1}{2} = -\frac{1}{6} + \frac{2}{3} \). Convert to sixths: \( \frac{3}{6} = -\frac{1}{6} + \frac{4}{6} \), and \( -\frac{1}{6} + \frac{4}{6} = \frac{3}{6} = \frac{1}{2} \), which is true.

Equation 12: \( p - 3\frac{1}{6} = -2\frac{1}{2} \)

Step1: Isolate \( p \)

Add \( 3\frac{1}{6} \) to both sides: \( p - 3\frac{1}{6} + 3\frac{1}{6} = -2\frac{1}{2} + 3\frac{1}{6} \)

Step2: Convert mixed numbers to improper fractions

\( -2\frac{1}{2} = -\frac{5}{2} \), \( 3\frac{1}{6} = \frac{19}{6} \), \( -\frac{5}{2} = -\frac{15}{6} \)
So, \( p = -\frac{15}{6} + \frac{19}{6} \)

Step3: Add the fractions

\( p = \frac{-15 + 19}{6} = \frac{4}{6} = \frac{2}{3} \)

Check:

Substitute \( p = \frac{2}{3} \) into the original equation: \( \frac{2}{3} - 3\frac{1}{6} = -2\frac{1}{2} \). Convert to sixths: \( \frac{4}{6} - \frac{19}{6} = -\frac{15}{6} = -2\frac{1}{2} \), which is true.

Equation 13: \( g - 9 = -19 \)

Step1: Isolate \( g \)

Add 9 to both sides: \( g - 9 + 9 = -19 + 9 \)

Step2: Simplify

\( g = -10 \)

Check:

Substitute \( g = -10 \) into the original equation: \( -10 - 9 = -19 \), which is true.

Equation 14: \( -9.3 = d - 3.4 \)

Step1: Isolate \( d \)

Add 3.4 to both sides: \( -9.3 + 3.4 = d - 3.4 + 3.4 \)

Step2: Simplify

\( -5.9 = d \)

Check:

Substitute \( d = -5.9 \) into the original equation: \( -9.3 = -5.9 - 3.4 \), and \( -5.9 - 3.4 = -9.3 \), which is true.

Equation 15: \( 4.58 + y = 2.5 \)

Step1: Isolate \( y \)

Subtract 4.58 from both sides: \( 4.58 + y - 4.58 = 2.5 - 4.58 \)

Step2: Simplify

\( y = -2.08 \)

Check:

Substitute \( y = -2.08 \) into the original equation: \( 4.58 + (-2.08) = 2.5 \), and \( 4.58 - 2.08 = 2.5 \), which is true.

Equation 16: \( x - 5.2 = -18.73 \)

Step1: Isolate \( x \)

Add 5.2 to both sides: \( x - 5.2 + 5.2 = -18.73 + 5.2 \)

Step2: Simplify

\( x = -13.53 \)

Check:

Substitute \( x = -13.53 \) into the original equation: \( -13.53 - 5.2 = -18.73 \), and \( -13.53 - 5.2 = -18.73 \), which is true.

Equation 17: \( q + \frac{5}{9} = \frac{1}{6} \)

Step1: Isolate \( q \)

Subtract \( \frac{5}{9} \) from both sides: \( q + \frac{5}{9} - \frac{5}{9} = \frac{1}{6} - \frac{5}{9} \)

Step2: Find a common denominator (18)

\( \frac{1}{6} = \frac{3}{18} \), \( \frac{5}{9} = \frac{10}{18} \)
So, \( q = \frac{3}{18} - \frac{10}{18} \)

Step3: Subtract the fractions

\( q = \frac{3 - 10}{18} = -\frac{7}{18} \)

Check:

Substitute \( q = -\frac{7}{18} \) into the original equation: \( -\frac{7}{18} + \frac{5}{9} = \frac{1}{6} \). Convert \( \frac{5}{9} \) to eighteenths: \( \frac{10}{18} \), so \( -\frac{7}{18} + \frac{10}{18} = \frac{3}{18} = \frac{1}{6} \), which is true.

Equation 18: \( -2\frac{1}{4} = r - \frac{4}{5} \)

Step1: Isolate \( r \)

Add \( \frac{4}{5} \) to both sides: \( -2\frac{1}{4} + \frac{4}{5} = r - \frac{4}{5} + \frac{4}{5} \)

Step2: Convert mixed number to improper fraction and find common denominator (20)

\( -2\frac{1}{4} = -\frac{9}{4} = -\frac{45}{20} \), \( \frac{4}{5} = \frac{16}{20} \)
So, \( r = -\frac{45}{20} + \frac{16}{20} \)

Step3: Add the fractions

\( r = \frac{-45 + 16}{20} = -\frac{29}{20} = -1\frac{9}{20} \)

Check:

Substitute \( r = -1\frac{9}{20} \) into the original equation: \( -2\frac{1}{4} = -1\frac{9}{20} - \frac{4}{5} \). Convert to twentieths: \( -2\frac{5}{20} = -1\frac{9}{20} - \frac{16}{20} \), \( -1\frac{9}{20} - \frac{16}{20} = -1\frac{25}{20} = -2\frac{5}{20} = -2\frac{1}{4} \), which is true.

Equation 19: \( w + 3\frac{3}{8} = 1\frac{5}{6} \)

Step1: Isolate \( w \)

Subtract \( 3\frac{3}{8} \) from both sides: \( w + 3\frac{3}{8} - 3\frac{3}{8} = 1\frac{5}{6} - 3\frac{3}{8} \)

Step2: Convert mixed numbers to improper fractions and find common denominator (24)

\( 1\frac{5}{6} = \frac{11}{6} = \frac{44}{24} \), \( 3\frac{3}{8} = \frac{27}{8} = \frac{81}{24} \)
So, \( w = \frac{44}{24} - \frac{81}{24} \)

Step3: Subtract the fractions

\( w = \frac{44 - 81}{24} = -\frac{37}{24} = -1\frac{13}{24} \)

Check:

Substitute \( w = -1\frac{13}{24} \) into the original equation: \( -1\frac{13}{24} + 3\frac{3}{8} = 1\frac{5}{6} \). Convert to twenty - fourths: \( -1\frac{13}{24} + 3\frac{9}{24} = 2\frac{22}{24} = 2\frac{11}{12} \)? Wait, no, let's recalculate. Wait, \( 3\frac{3}{8}=3 + \frac{3}{8}=\frac{24 + 3}{8}=\frac{27}{8}=\frac{81}{24} \), \( -1\frac{13}{24}=-\frac{37}{24} \), \( -\frac{37}{24}+\frac{81}{24}=\frac{44}{24}=\frac{11}{6}=1\frac{5}{6} \), which is correct.

Equation 20: \( 4\frac{2}{5}+k=-3\frac{2}{11} \)

Step1: Isolate \( k \)

Subtract \( 4\frac{2}{5} \) from both sides: \( k=-3\frac{2}{11}-4\frac{2}{5} \)

Step2: Convert mixed numbers to improper fractions and find common denominator (55)

\( -3\frac{2}{11}=-\frac{35}{11}=-\frac{175}{55} \), \( 4\frac{2}{5}=\frac{22}{5}=\frac{242}{55} \)
So, \( k = -\frac{175}{55}-\frac{242}{55} \)

Step3: Subtract the fractions

\( k=\frac{-175 - 242}{55}=-\frac{417}{55}=-7\frac{32}{55} \)

Check:

Substitute \( k = -7\frac{32}{55} \) into the original equation: \( 4\frac{2}{5}+(-7\frac{32}{55})=-3\frac{2}{11} \). Convert to fifty - fifths: \( 4\frac{22}{55}-7\frac{32}{55}=-3\frac{10}{55}=-3\frac{2}{11} \), which is true.

Answers:

5. \( a = \boldsymbol{19} \)
6. \( z=\boldsymbol{5} \)
7. \( k = \boldsymbol{-20} \)
8. \( x=\boldsymbol{-18} \)
9. \( c=\boldsymbol{3.6} \)
10. \( w=\boldsymbol{-15.4} \)
11. \( q=\boldsymbol{-\frac{1}{6}} \)
12. \( p=\boldsymbol{\frac{2}{3}} \)
13. \( g=\boldsymbol{-10} \)
14. \( d=\boldsymbol{-5.9} \)
15. \( y=\boldsymbol{-2.08} \)
16. \( x=\boldsymbol{-13.53} \)
17. \( q=\boldsymbol{-\frac{7}{18}} \)
18. \( r=\boldsymbol{-1\frac{9}{20}} \)
19. \( w=\boldsymbol{-1\frac{13}{24}} \)
20. \( k=\boldsymbol{-7\frac{32}{55}} \)

Answer:

s:

1. \( a = \boldsymbol{19} \)
2. \( z=\boldsymbol{5} \)
3. \( k = \boldsymbol{-20} \)
4. \( x=\boldsymbol{-18} \)
5. \( c=\boldsymbol{3.6} \)
6. \( w=\boldsymbol{-15.4} \)
7. \( q=\boldsymbol{-\frac{1}{6}} \)
8. \( p=\boldsymbol{\frac{2}{3}} \)
9. \( g=\boldsymbol{-10} \)
10. \( d=\boldsymbol{-5.9} \)
11. \( y=\boldsymbol{-2.08} \)
12. \( x=\boldsymbol{-13.53} \)
13. \( q=\boldsymbol{-\frac{7}{18}} \)
14. \( r=\boldsymbol{-1\frac{9}{20}} \)
15. \( w=\boldsymbol{-1\frac{13}{24}} \)
16. \( k=\boldsymbol{-7\frac{32}{55}} \)