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To determine which linear equations have an infinite number of solutions, we need to check whether each equation simplifies to an identity. An identity is true for all values of the variable, indicating infinite solutions. We'll simplify each equation step by step.
### Equation 1
[tex]\[ \left(x - \frac{3}{7}\right) = \frac{2}{3}\left(\frac{3}{2} x - \frac{9}{14}\right) \][/tex]
First, simplify the right-hand side:
[tex]\[ \frac{2}{3} \left(\frac{3}{2} x - \frac{9}{14}\right) = \frac{2}{3} \cdot \frac{3}{2} x - \frac{2}{3} \cdot \frac{9}{14} = x - \frac{3}{7} \][/tex]
So the equation becomes:
[tex]\[ x - \frac{3}{7} = x - \frac{3}{7} \][/tex]
This is an identity since both sides are equal for all values of [tex]\(x\)[/tex]. Therefore, this equation has an infinite number of solutions.
### Equation 2
[tex]\[ 8(x + 2) = 5x - 14 \][/tex]
Expand and simplify:
[tex]\[ 8x + 16 = 5x - 14 \][/tex]
Subtract [tex]\(5x\)[/tex] from both sides:
[tex]\[ 3x + 16 = -14 \][/tex]
Subtract 16 from both sides:
[tex]\[ 3x = -30 \][/tex]
Divide by 3:
[tex]\[ x = -10 \][/tex]
This equation has a unique solution [tex]\(x = -10\)[/tex], so there are not an infinite number of solutions.
### Equation 3
[tex]\[ 12.3x - 18 = 3(-6 + 4.1x) \][/tex]
Expand the right-hand side:
[tex]\[ 12.3x - 18 = 3(-6) + 3(4.1x) = -18 + 12.3x \][/tex]
So the equation becomes:
[tex]\[ 12.3x - 18 = 12.3x - 18 \][/tex]
This is an identity since both sides are equal for all values of [tex]\(x\)[/tex]. Therefore, this equation has an infinite number of solutions.
### Equation 4
[tex]\[ \frac{1}{2}(6x + 10) = 7\left(\frac{3}{7}x - 2\right) \][/tex]
Simplify both sides:
[tex]\[ \frac{1}{2} \cdot 6x + \frac{1}{2} \cdot 10 = 3x - 14 \][/tex]
[tex]\[ 3x + 5 = 3x - 14 \][/tex]
Subtract [tex]\(3x\)[/tex] from both sides:
[tex]\[ 5 \neq -14 \][/tex]
This is a contradiction, so the equation has no solution and definitely not an infinite number of solutions.
### Equation 5
[tex]\[ 4.2x - 3.5 = 2.1(5x + 8) \][/tex]
Expand and simplify:
[tex]\[ 4.2x - 3.5 = 2.1 \cdot 5x + 2.1 \cdot 8 = 10.5x + 16.8 \][/tex]
Bring all terms involving [tex]\(x\)[/tex] on one side and constants on the other side:
[tex]\[ 4.2x - 10.5x = 16.8 + 3.5 \][/tex]
[tex]\[ -6.3x = 20.3 \][/tex]
Divide by [tex]\(-6.3\)[/tex]:
[tex]\[ x = -\frac{20.3}{6.3} \][/tex]
This gives a unique solution, not an infinite number of solutions.
### Conclusion
The equations that have an infinite number of solutions are:
- [tex]\(\left(x - \frac{3}{7}\right) = \frac{2}{3}\left(\frac{3}{2}x - \frac{9}{14}\right)\)[/tex]
- [tex]\(12.3x - 18 = 3(-6 + 4.1x)\)[/tex]
### Equation 1
[tex]\[ \left(x - \frac{3}{7}\right) = \frac{2}{3}\left(\frac{3}{2} x - \frac{9}{14}\right) \][/tex]
First, simplify the right-hand side:
[tex]\[ \frac{2}{3} \left(\frac{3}{2} x - \frac{9}{14}\right) = \frac{2}{3} \cdot \frac{3}{2} x - \frac{2}{3} \cdot \frac{9}{14} = x - \frac{3}{7} \][/tex]
So the equation becomes:
[tex]\[ x - \frac{3}{7} = x - \frac{3}{7} \][/tex]
This is an identity since both sides are equal for all values of [tex]\(x\)[/tex]. Therefore, this equation has an infinite number of solutions.
### Equation 2
[tex]\[ 8(x + 2) = 5x - 14 \][/tex]
Expand and simplify:
[tex]\[ 8x + 16 = 5x - 14 \][/tex]
Subtract [tex]\(5x\)[/tex] from both sides:
[tex]\[ 3x + 16 = -14 \][/tex]
Subtract 16 from both sides:
[tex]\[ 3x = -30 \][/tex]
Divide by 3:
[tex]\[ x = -10 \][/tex]
This equation has a unique solution [tex]\(x = -10\)[/tex], so there are not an infinite number of solutions.
### Equation 3
[tex]\[ 12.3x - 18 = 3(-6 + 4.1x) \][/tex]
Expand the right-hand side:
[tex]\[ 12.3x - 18 = 3(-6) + 3(4.1x) = -18 + 12.3x \][/tex]
So the equation becomes:
[tex]\[ 12.3x - 18 = 12.3x - 18 \][/tex]
This is an identity since both sides are equal for all values of [tex]\(x\)[/tex]. Therefore, this equation has an infinite number of solutions.
### Equation 4
[tex]\[ \frac{1}{2}(6x + 10) = 7\left(\frac{3}{7}x - 2\right) \][/tex]
Simplify both sides:
[tex]\[ \frac{1}{2} \cdot 6x + \frac{1}{2} \cdot 10 = 3x - 14 \][/tex]
[tex]\[ 3x + 5 = 3x - 14 \][/tex]
Subtract [tex]\(3x\)[/tex] from both sides:
[tex]\[ 5 \neq -14 \][/tex]
This is a contradiction, so the equation has no solution and definitely not an infinite number of solutions.
### Equation 5
[tex]\[ 4.2x - 3.5 = 2.1(5x + 8) \][/tex]
Expand and simplify:
[tex]\[ 4.2x - 3.5 = 2.1 \cdot 5x + 2.1 \cdot 8 = 10.5x + 16.8 \][/tex]
Bring all terms involving [tex]\(x\)[/tex] on one side and constants on the other side:
[tex]\[ 4.2x - 10.5x = 16.8 + 3.5 \][/tex]
[tex]\[ -6.3x = 20.3 \][/tex]
Divide by [tex]\(-6.3\)[/tex]:
[tex]\[ x = -\frac{20.3}{6.3} \][/tex]
This gives a unique solution, not an infinite number of solutions.
### Conclusion
The equations that have an infinite number of solutions are:
- [tex]\(\left(x - \frac{3}{7}\right) = \frac{2}{3}\left(\frac{3}{2}x - \frac{9}{14}\right)\)[/tex]
- [tex]\(12.3x - 18 = 3(-6 + 4.1x)\)[/tex]
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