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Sagot :
Sure, let's solve these equations in detail step-by-step.
### (g) [tex]\(3a - 2.4 = 3.6\)[/tex]
Step 1: Add 2.4 to both sides.
[tex]\[ 3a = 3.6 + 2.4 \][/tex]
Step 2: Simplify.
[tex]\[ 3a = 6 \][/tex]
Step 3: Divide both sides by 3.
[tex]\[ a = 2 \][/tex]
So, the solution is [tex]\( a = 2.0 \)[/tex].
### (h) [tex]\(2.1a + 5.3 = 13.7\)[/tex]
Step 1: Subtract 5.3 from both sides.
[tex]\[ 2.1a = 13.7 - 5.3 \][/tex]
Step 2: Simplify.
[tex]\[ 2.1a = 8.4 \][/tex]
Step 3: Divide both sides by 2.1.
[tex]\[ a = 4 \][/tex]
So, the solution is [tex]\( a = 3.999999999999999 \)[/tex].
### 3. Solve the following equations:
#### (a) [tex]\(\frac{a}{2} - 3 = 4\)[/tex]
Step 1: Add 3 to both sides.
[tex]\[ \frac{a}{2} = 4 + 3 \][/tex]
Step 2: Simplify.
[tex]\[ \frac{a}{2} = 7 \][/tex]
Step 3: Multiply both sides by 2.
[tex]\[ a = 14 \][/tex]
So, the solution is [tex]\( a = 14 \)[/tex].
#### (b) [tex]\(\frac{x}{3} + 5 = 8\)[/tex]
Step 1: Subtract 5 from both sides.
[tex]\[ \frac{x}{3} = 8 - 5 \][/tex]
Step 2: Simplify.
[tex]\[ \frac{x}{3} = 3 \][/tex]
Step 3: Multiply both sides by 3.
[tex]\[ x = 9 \][/tex]
So, the solution is [tex]\( x = 9 \)[/tex].
#### (c) [tex]\(\frac{b}{2} + 2 \frac{1}{3} = 5\)[/tex]
First, convert the mixed number [tex]\(2 \frac{1}{3}\)[/tex] to an improper fraction.
[tex]\[ 2 \frac{1}{3} = \frac{7}{3} \][/tex]
Step 1: Subtract [tex]\(\frac{7}{3}\)[/tex] from both sides.
[tex]\[ \frac{b}{2} = 5 - \frac{7}{3} \][/tex]
To subtract, convert 5 to [tex]\(\frac{15}{3}\)[/tex].
[tex]\[ \frac{b}{2} = \frac{15}{3} - \frac{7}{3} \][/tex]
Step 2: Simplify.
[tex]\[ \frac{b}{2} = \frac{8}{3} \][/tex]
Step 3: Multiply both sides by 2.
[tex]\[ b = 2 \times \frac{8}{3} = \frac{16}{3} \approx 5.333 \][/tex]
So, the solution is [tex]\( b = 5.333333333333333 \)[/tex].
#### (d) [tex]\(\frac{x}{5} - 4 = 2.4\)[/tex]
Step 1: Add 4 to both sides.
[tex]\[ \frac{x}{5} = 2.4 + 4 \][/tex]
Step 2: Simplify.
[tex]\[ \frac{x}{5} = 6.4 \][/tex]
Step 3: Multiply both sides by 5.
[tex]\[ x = 6.4 \times 5 = 32 \][/tex]
So, the solution is [tex]\( x = 32 \)[/tex].
#### (e) [tex]\(\frac{2}{5} x - 3 = 5\)[/tex]
Step 1: Add 3 to both sides.
[tex]\[ \frac{2}{5} x = 5 + 3 \][/tex]
Step 2: Simplify.
[tex]\[ \frac{2}{5} x = 8 \][/tex]
Step 3: Multiply both sides by [tex]\(\frac{5}{2}\)[/tex].
[tex]\[ x = 8 \times \frac{5}{2} = 20 \][/tex]
So, the solution is [tex]\( x = 20 \)[/tex].
#### (f) [tex]\(\frac{z}{1.5} + 4.2 = 13.2\)[/tex]
Step 1: Subtract 4.2 from both sides.
[tex]\[ \frac{z}{1.5} = 13.2 - 4.2 \][/tex]
Step 2: Simplify.
[tex]\[ \frac{z}{1.5} = 9 \][/tex]
Step 3: Multiply both sides by 1.5.
[tex]\[ z = 9 \times 1.5 = 13.5 \][/tex]
So, the solution is [tex]\( z = 13.5 \)[/tex].
#### (g) [tex]\(\frac{3}{4} x + 1 = 1 \frac{7}{8}\)[/tex]
First, convert the mixed number [tex]\(1 \frac{7}{8}\)[/tex] to an improper fraction.
[tex]\[ 1 \frac{7}{8} = \frac{15}{8} \][/tex]
Step 1: Subtract 1 from both sides.
[tex]\[ \frac{3}{4} x = \frac{15}{8} - 1 \][/tex]
Convert 1 to [tex]\(\frac{8}{8}\)[/tex].
[tex]\[ \frac{3}{4} x = \frac{15}{8} - \frac{8}{8} \][/tex]
Step 2: Simplify.
[tex]\[ \frac{3}{4} x = \frac{7}{8} \][/tex]
Step 3: Multiply both sides by [tex]\(\frac{4}{3}\)[/tex].
[tex]\[ x = \frac{7}{8} \times \frac{4}{3} = \frac{28}{24} = \frac{7}{6} \][/tex]
So, the solution is [tex]\( x = 1.1666666666666667 \)[/tex].
### (g) [tex]\(3a - 2.4 = 3.6\)[/tex]
Step 1: Add 2.4 to both sides.
[tex]\[ 3a = 3.6 + 2.4 \][/tex]
Step 2: Simplify.
[tex]\[ 3a = 6 \][/tex]
Step 3: Divide both sides by 3.
[tex]\[ a = 2 \][/tex]
So, the solution is [tex]\( a = 2.0 \)[/tex].
### (h) [tex]\(2.1a + 5.3 = 13.7\)[/tex]
Step 1: Subtract 5.3 from both sides.
[tex]\[ 2.1a = 13.7 - 5.3 \][/tex]
Step 2: Simplify.
[tex]\[ 2.1a = 8.4 \][/tex]
Step 3: Divide both sides by 2.1.
[tex]\[ a = 4 \][/tex]
So, the solution is [tex]\( a = 3.999999999999999 \)[/tex].
### 3. Solve the following equations:
#### (a) [tex]\(\frac{a}{2} - 3 = 4\)[/tex]
Step 1: Add 3 to both sides.
[tex]\[ \frac{a}{2} = 4 + 3 \][/tex]
Step 2: Simplify.
[tex]\[ \frac{a}{2} = 7 \][/tex]
Step 3: Multiply both sides by 2.
[tex]\[ a = 14 \][/tex]
So, the solution is [tex]\( a = 14 \)[/tex].
#### (b) [tex]\(\frac{x}{3} + 5 = 8\)[/tex]
Step 1: Subtract 5 from both sides.
[tex]\[ \frac{x}{3} = 8 - 5 \][/tex]
Step 2: Simplify.
[tex]\[ \frac{x}{3} = 3 \][/tex]
Step 3: Multiply both sides by 3.
[tex]\[ x = 9 \][/tex]
So, the solution is [tex]\( x = 9 \)[/tex].
#### (c) [tex]\(\frac{b}{2} + 2 \frac{1}{3} = 5\)[/tex]
First, convert the mixed number [tex]\(2 \frac{1}{3}\)[/tex] to an improper fraction.
[tex]\[ 2 \frac{1}{3} = \frac{7}{3} \][/tex]
Step 1: Subtract [tex]\(\frac{7}{3}\)[/tex] from both sides.
[tex]\[ \frac{b}{2} = 5 - \frac{7}{3} \][/tex]
To subtract, convert 5 to [tex]\(\frac{15}{3}\)[/tex].
[tex]\[ \frac{b}{2} = \frac{15}{3} - \frac{7}{3} \][/tex]
Step 2: Simplify.
[tex]\[ \frac{b}{2} = \frac{8}{3} \][/tex]
Step 3: Multiply both sides by 2.
[tex]\[ b = 2 \times \frac{8}{3} = \frac{16}{3} \approx 5.333 \][/tex]
So, the solution is [tex]\( b = 5.333333333333333 \)[/tex].
#### (d) [tex]\(\frac{x}{5} - 4 = 2.4\)[/tex]
Step 1: Add 4 to both sides.
[tex]\[ \frac{x}{5} = 2.4 + 4 \][/tex]
Step 2: Simplify.
[tex]\[ \frac{x}{5} = 6.4 \][/tex]
Step 3: Multiply both sides by 5.
[tex]\[ x = 6.4 \times 5 = 32 \][/tex]
So, the solution is [tex]\( x = 32 \)[/tex].
#### (e) [tex]\(\frac{2}{5} x - 3 = 5\)[/tex]
Step 1: Add 3 to both sides.
[tex]\[ \frac{2}{5} x = 5 + 3 \][/tex]
Step 2: Simplify.
[tex]\[ \frac{2}{5} x = 8 \][/tex]
Step 3: Multiply both sides by [tex]\(\frac{5}{2}\)[/tex].
[tex]\[ x = 8 \times \frac{5}{2} = 20 \][/tex]
So, the solution is [tex]\( x = 20 \)[/tex].
#### (f) [tex]\(\frac{z}{1.5} + 4.2 = 13.2\)[/tex]
Step 1: Subtract 4.2 from both sides.
[tex]\[ \frac{z}{1.5} = 13.2 - 4.2 \][/tex]
Step 2: Simplify.
[tex]\[ \frac{z}{1.5} = 9 \][/tex]
Step 3: Multiply both sides by 1.5.
[tex]\[ z = 9 \times 1.5 = 13.5 \][/tex]
So, the solution is [tex]\( z = 13.5 \)[/tex].
#### (g) [tex]\(\frac{3}{4} x + 1 = 1 \frac{7}{8}\)[/tex]
First, convert the mixed number [tex]\(1 \frac{7}{8}\)[/tex] to an improper fraction.
[tex]\[ 1 \frac{7}{8} = \frac{15}{8} \][/tex]
Step 1: Subtract 1 from both sides.
[tex]\[ \frac{3}{4} x = \frac{15}{8} - 1 \][/tex]
Convert 1 to [tex]\(\frac{8}{8}\)[/tex].
[tex]\[ \frac{3}{4} x = \frac{15}{8} - \frac{8}{8} \][/tex]
Step 2: Simplify.
[tex]\[ \frac{3}{4} x = \frac{7}{8} \][/tex]
Step 3: Multiply both sides by [tex]\(\frac{4}{3}\)[/tex].
[tex]\[ x = \frac{7}{8} \times \frac{4}{3} = \frac{28}{24} = \frac{7}{6} \][/tex]
So, the solution is [tex]\( x = 1.1666666666666667 \)[/tex].
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