Answer :
To identify which equation results in a different value of [tex]\( x \)[/tex] compared to the other three, we need to solve each of the given equations and compare their solutions.
Let's go through each equation.
1. For the equation:
[tex]\[ -\frac{7}{8} x - \frac{3}{4} = 20 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ -\frac{7}{8} x = 20 + \frac{3}{4} \][/tex]
[tex]\[ -\frac{7}{8} x = \frac{80}{4} + \frac{3}{4} \][/tex]
[tex]\[ -\frac{7}{8} x = \frac{83}{4} \][/tex]
[tex]\[ x = -\frac{83}{4} \cdot \frac{8}{7} \][/tex]
[tex]\[ x = -\frac{83 \cdot 2}{7} \][/tex]
[tex]\[ x = -\frac{166}{7} \approx -23.714 \][/tex]
2. For the equation:
[tex]\[ \frac{3}{4} + \frac{7}{8} x = -20 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ \frac{7}{8} x = -20 - \frac{3}{4} \][/tex]
[tex]\[ \frac{7}{8} x = -\frac{80}{4} - \frac{3}{4} \][/tex]
[tex]\[ \frac{7}{8} x = -\frac{83}{4} \][/tex]
[tex]\[ x = -\frac{83}{4} \cdot \frac{8}{7} \][/tex]
[tex]\[ x = -\frac{83 \cdot 2}{7} \][/tex]
[tex]\[ x = -\frac{166}{7} \approx -23.714 \][/tex]
3. For the equation:
[tex]\[ -7 \left(\frac{1}{8}\right) x - \frac{3}{4} = 20 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ -\frac{7}{8} x - \frac{3}{4} = 20 \][/tex]
Recognizing this is the same as the first equation:
[tex]\[ x = -23.714 \][/tex]
4. For the equation:
[tex]\[ -\frac{7}{8} \left( -\frac{8}{7} \right) x - \frac{3}{4} = 20 \left( -\frac{8}{7} \right) \][/tex]
Simplifying the left-hand side:
[tex]\[ x - \frac{3}{4} = -\frac{160}{7} \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ x = -\frac{160}{7} + \frac{3}{4} \][/tex]
Converting [tex]\(\frac{3}{4}\)[/tex] to a common denominator:
[tex]\[ \frac{3}{4} = \frac{3 \cdot 7}{4 \cdot 7} = \frac{21}{28} = \frac{21 \cdot 4}{28} = \frac{21 \cdot 4}{28 / 4} = \frac{84}{28} \][/tex]
[tex]\[ x = -\frac{160 \cdot 4}{7 \cdot 4} \cdot + \frac{3 \cdot 7}{4 \cdot 7} \neq -\frac{160 \cdot 4}{28 \cdot 4}\approx -22.107 \][/tex]
Comparing the solutions:
1. [tex]\( x \approx -23.714 \)[/tex]
2. [tex]\( x \approx -23.714 \)[/tex]
3. [tex]\( x \approx -23.714 \)[/tex]
4. [tex]\( x \approx -22.107 \)[/tex]
The equation resulting in a different value of [tex]\( x \)[/tex] is:
[tex]\[ -\frac{7}{8} \left( -\frac{8}{7} \right) x - \frac{3}{4} = 20 \left( -\frac{8}{7} \right) \][/tex]
Therefore, the equation
[tex]\[ -\frac{7}{8} \left( -\frac{8}{7} \right) x - \frac{3}{4} = 20 \left( -\frac{8}{7} \right) \][/tex]
results in a different value of [tex]\( x \)[/tex] than the other three equations.
Let's go through each equation.
1. For the equation:
[tex]\[ -\frac{7}{8} x - \frac{3}{4} = 20 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ -\frac{7}{8} x = 20 + \frac{3}{4} \][/tex]
[tex]\[ -\frac{7}{8} x = \frac{80}{4} + \frac{3}{4} \][/tex]
[tex]\[ -\frac{7}{8} x = \frac{83}{4} \][/tex]
[tex]\[ x = -\frac{83}{4} \cdot \frac{8}{7} \][/tex]
[tex]\[ x = -\frac{83 \cdot 2}{7} \][/tex]
[tex]\[ x = -\frac{166}{7} \approx -23.714 \][/tex]
2. For the equation:
[tex]\[ \frac{3}{4} + \frac{7}{8} x = -20 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ \frac{7}{8} x = -20 - \frac{3}{4} \][/tex]
[tex]\[ \frac{7}{8} x = -\frac{80}{4} - \frac{3}{4} \][/tex]
[tex]\[ \frac{7}{8} x = -\frac{83}{4} \][/tex]
[tex]\[ x = -\frac{83}{4} \cdot \frac{8}{7} \][/tex]
[tex]\[ x = -\frac{83 \cdot 2}{7} \][/tex]
[tex]\[ x = -\frac{166}{7} \approx -23.714 \][/tex]
3. For the equation:
[tex]\[ -7 \left(\frac{1}{8}\right) x - \frac{3}{4} = 20 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ -\frac{7}{8} x - \frac{3}{4} = 20 \][/tex]
Recognizing this is the same as the first equation:
[tex]\[ x = -23.714 \][/tex]
4. For the equation:
[tex]\[ -\frac{7}{8} \left( -\frac{8}{7} \right) x - \frac{3}{4} = 20 \left( -\frac{8}{7} \right) \][/tex]
Simplifying the left-hand side:
[tex]\[ x - \frac{3}{4} = -\frac{160}{7} \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ x = -\frac{160}{7} + \frac{3}{4} \][/tex]
Converting [tex]\(\frac{3}{4}\)[/tex] to a common denominator:
[tex]\[ \frac{3}{4} = \frac{3 \cdot 7}{4 \cdot 7} = \frac{21}{28} = \frac{21 \cdot 4}{28} = \frac{21 \cdot 4}{28 / 4} = \frac{84}{28} \][/tex]
[tex]\[ x = -\frac{160 \cdot 4}{7 \cdot 4} \cdot + \frac{3 \cdot 7}{4 \cdot 7} \neq -\frac{160 \cdot 4}{28 \cdot 4}\approx -22.107 \][/tex]
Comparing the solutions:
1. [tex]\( x \approx -23.714 \)[/tex]
2. [tex]\( x \approx -23.714 \)[/tex]
3. [tex]\( x \approx -23.714 \)[/tex]
4. [tex]\( x \approx -22.107 \)[/tex]
The equation resulting in a different value of [tex]\( x \)[/tex] is:
[tex]\[ -\frac{7}{8} \left( -\frac{8}{7} \right) x - \frac{3}{4} = 20 \left( -\frac{8}{7} \right) \][/tex]
Therefore, the equation
[tex]\[ -\frac{7}{8} \left( -\frac{8}{7} \right) x - \frac{3}{4} = 20 \left( -\frac{8}{7} \right) \][/tex]
results in a different value of [tex]\( x \)[/tex] than the other three equations.
