Answer :
Certainly! Let's carefully solve the equations step-by-step to understand how we would arrive at the solution.
### First Equation:
[tex]\[ 4(3x + 5) - 3 = 9x - 7 \][/tex]
1. Distribute the 4 on the left side:
[tex]\[ 4 \times 3x + 4 \times 5 - 3 = 9x - 7 \][/tex]
[tex]\[ 12x + 20 - 3 = 9x - 7 \][/tex]
2. Simplify the left side:
[tex]\[ 12x + 17 = 9x - 7 \][/tex]
3. Move the [tex]\(9x\)[/tex] term to the left side by subtracting [tex]\(9x\)[/tex] from both sides:
[tex]\[ 12x + 17 - 9x = -7 \][/tex]
[tex]\[ 3x + 17 = -7 \][/tex]
4. Move the 17 to the right side by subtracting 17 from both sides:
[tex]\[ 3x = -7 - 17 \][/tex]
[tex]\[ 3x = -24 \][/tex]
5. Divide both sides by 3:
[tex]\[ x = -8 \][/tex]
Thus, [tex]\(x = -8\)[/tex].
### Second Equation:
[tex]\[ \frac{1}{3}(5x - 9) = 2\left(\frac{1}{3}x + 6\right) \][/tex]
1. Distribute [tex]\(\frac{1}{3}\)[/tex] on the left side and distribute [tex]\(2\)[/tex] on the right side:
[tex]\[ \frac{1}{3} \times 5x - \frac{1}{3} \times 9 = 2 \times \frac{1}{3}x + 2 \times 6 \][/tex]
[tex]\[ \frac{5x}{3} - 3 = \frac{2x}{3} + 12 \][/tex]
2. Clear the fractions by multiplying every term by 3:
[tex]\[ 3 \times \left(\frac{5x}{3}\right) - 3 \times 3 = 3 \times \left(\frac{2x}{3}\right) + 3 \times 12 \][/tex]
[tex]\[ 5x - 9 = 2x + 36 \][/tex]
3. Move the [tex]\(2x\)[/tex] term to the left side by subtracting [tex]\(2x\)[/tex] from both sides:
[tex]\[ 5x - 2x - 9 = 36 \][/tex]
[tex]\[ 3x - 9 = 36 \][/tex]
4. Move the [tex]\(-9\)[/tex] to the right side by adding 9 to both sides:
[tex]\[ 3x = 36 + 9 \][/tex]
[tex]\[ 3x = 45 \][/tex]
5. Divide both sides by 3:
[tex]\[ x = 15 \][/tex]
Thus, [tex]\(x = 15\)[/tex].
### Third Equation:
[tex]\[ 5(x + 7) - 3(x - 4) = 7x + 2 \][/tex]
1. Distribute the 5 and (-3) on the left side:
[tex]\[ 5 \times x + 5 \times 7 - 3 \times x + 3 \times 4 = 7x + 2 \][/tex]
[tex]\[ 5x + 35 - 3x + 12 = 7x + 2 \][/tex]
2. Combine like terms on the left side:
[tex]\[ (5x - 3x) + (35 + 12) = 7x + 2 \][/tex]
[tex]\[ 2x + 47 = 7x + 2 \][/tex]
3. Move the [tex]\(7x\)[/tex] term to the left side by subtracting [tex]\(7x\)[/tex] from both sides:
[tex]\[ 2x - 7x + 47 = 2 \][/tex]
[tex]\[ -5x + 47 = 2 \][/tex]
4. Move the 47 to the right side by subtracting 47 from both sides:
[tex]\[ -5x = 2 - 47 \][/tex]
[tex]\[ -5x = -45 \][/tex]
5. Divide both sides by -5:
[tex]\[ x = \frac{-45}{-5} \][/tex]
[tex]\[ x = 9 \][/tex]
Thus, [tex]\(x = 9\)[/tex].
Our final solutions for the given equations are:
1. [tex]\(x = -8\)[/tex]
2. [tex]\(x = 15\)[/tex]
3. [tex]\(x = 9\)[/tex]
Now let's verify which [tex]\(x\)[/tex] values are correct based on our understanding:
Reset Next doesn't specify a final result, but if we consider that we are summarizing our findings, we get:
- The first equation yields [tex]\(x = -8\)[/tex],
- The second equation yields [tex]\(x = 15\)[/tex],
- The third equation yields [tex]\(x = 9\)[/tex].
And these steps align with our computed solutions.
### First Equation:
[tex]\[ 4(3x + 5) - 3 = 9x - 7 \][/tex]
1. Distribute the 4 on the left side:
[tex]\[ 4 \times 3x + 4 \times 5 - 3 = 9x - 7 \][/tex]
[tex]\[ 12x + 20 - 3 = 9x - 7 \][/tex]
2. Simplify the left side:
[tex]\[ 12x + 17 = 9x - 7 \][/tex]
3. Move the [tex]\(9x\)[/tex] term to the left side by subtracting [tex]\(9x\)[/tex] from both sides:
[tex]\[ 12x + 17 - 9x = -7 \][/tex]
[tex]\[ 3x + 17 = -7 \][/tex]
4. Move the 17 to the right side by subtracting 17 from both sides:
[tex]\[ 3x = -7 - 17 \][/tex]
[tex]\[ 3x = -24 \][/tex]
5. Divide both sides by 3:
[tex]\[ x = -8 \][/tex]
Thus, [tex]\(x = -8\)[/tex].
### Second Equation:
[tex]\[ \frac{1}{3}(5x - 9) = 2\left(\frac{1}{3}x + 6\right) \][/tex]
1. Distribute [tex]\(\frac{1}{3}\)[/tex] on the left side and distribute [tex]\(2\)[/tex] on the right side:
[tex]\[ \frac{1}{3} \times 5x - \frac{1}{3} \times 9 = 2 \times \frac{1}{3}x + 2 \times 6 \][/tex]
[tex]\[ \frac{5x}{3} - 3 = \frac{2x}{3} + 12 \][/tex]
2. Clear the fractions by multiplying every term by 3:
[tex]\[ 3 \times \left(\frac{5x}{3}\right) - 3 \times 3 = 3 \times \left(\frac{2x}{3}\right) + 3 \times 12 \][/tex]
[tex]\[ 5x - 9 = 2x + 36 \][/tex]
3. Move the [tex]\(2x\)[/tex] term to the left side by subtracting [tex]\(2x\)[/tex] from both sides:
[tex]\[ 5x - 2x - 9 = 36 \][/tex]
[tex]\[ 3x - 9 = 36 \][/tex]
4. Move the [tex]\(-9\)[/tex] to the right side by adding 9 to both sides:
[tex]\[ 3x = 36 + 9 \][/tex]
[tex]\[ 3x = 45 \][/tex]
5. Divide both sides by 3:
[tex]\[ x = 15 \][/tex]
Thus, [tex]\(x = 15\)[/tex].
### Third Equation:
[tex]\[ 5(x + 7) - 3(x - 4) = 7x + 2 \][/tex]
1. Distribute the 5 and (-3) on the left side:
[tex]\[ 5 \times x + 5 \times 7 - 3 \times x + 3 \times 4 = 7x + 2 \][/tex]
[tex]\[ 5x + 35 - 3x + 12 = 7x + 2 \][/tex]
2. Combine like terms on the left side:
[tex]\[ (5x - 3x) + (35 + 12) = 7x + 2 \][/tex]
[tex]\[ 2x + 47 = 7x + 2 \][/tex]
3. Move the [tex]\(7x\)[/tex] term to the left side by subtracting [tex]\(7x\)[/tex] from both sides:
[tex]\[ 2x - 7x + 47 = 2 \][/tex]
[tex]\[ -5x + 47 = 2 \][/tex]
4. Move the 47 to the right side by subtracting 47 from both sides:
[tex]\[ -5x = 2 - 47 \][/tex]
[tex]\[ -5x = -45 \][/tex]
5. Divide both sides by -5:
[tex]\[ x = \frac{-45}{-5} \][/tex]
[tex]\[ x = 9 \][/tex]
Thus, [tex]\(x = 9\)[/tex].
Our final solutions for the given equations are:
1. [tex]\(x = -8\)[/tex]
2. [tex]\(x = 15\)[/tex]
3. [tex]\(x = 9\)[/tex]
Now let's verify which [tex]\(x\)[/tex] values are correct based on our understanding:
Reset Next doesn't specify a final result, but if we consider that we are summarizing our findings, we get:
- The first equation yields [tex]\(x = -8\)[/tex],
- The second equation yields [tex]\(x = 15\)[/tex],
- The third equation yields [tex]\(x = 9\)[/tex].
And these steps align with our computed solutions.
