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Sagot :
To find the solution set for the given system of equations, we need to solve the following simultaneous equations:
[tex]\[ \begin{array}{l} \frac{1}{5} x-\frac{1}{4} y=3 \\ \frac{2}{3} x+\frac{1}{2} y=3 \end{array} \][/tex]
Step-by-Step Solution:
1. Write the equations in a standard form:
[tex]\[ \frac{1}{5} x - \frac{1}{4} y = 3 \][/tex]
[tex]\[ \frac{2}{3} x + \frac{1}{2} y = 3 \][/tex]
2. Multiply both sides of each equation to get rid of the fractions:
For the first equation:
[tex]\[ 20 \left(\frac{1}{5} x - \frac{1}{4} y = 3 \right) \][/tex]
[tex]\[ 4x - 5y = 60 \][/tex]
For the second equation:
[tex]\[ 6 \left(\frac{2}{3} x + \frac{1}{2} y = 3 \right) \][/tex]
[tex]\[ 4x + 3y = 18 \][/tex]
3. Rewrite the system of equations:
[tex]\[ \begin{array}{l} 4x - 5y = 60 \\ 4x + 3y = 18 \end{array} \][/tex]
4. Subtract the second equation from the first to eliminate [tex]\( x \)[/tex]:
[tex]\[ (4x - 5y) - (4x + 3y) = 60 - 18 \][/tex]
[tex]\[ -8y = 42 \][/tex]
5. Solve for [tex]\( y \)[/tex]:
[tex]\[ y = -\frac{42}{8} \][/tex]
[tex]\[ y = -5.25 \][/tex]
6. Substitute [tex]\( y = -5.25 \)[/tex] back into one of the original equations to solve for [tex]\( x \)[/tex]:
Substituting into [tex]\( 4x - 5y = 60 \)[/tex]:
[tex]\[ 4x - 5(-5.25) = 60 \][/tex]
[tex]\[ 4x + 26.25 = 60 \][/tex]
[tex]\[ 4x = 60 - 26.25 \][/tex]
[tex]\[ 4x = 33.75 \][/tex]
[tex]\[ x = \frac{33.75}{4} \][/tex]
[tex]\[ x = 8.4375 \][/tex]
7. The solution set for the system of equations is:
[tex]\[ \{ (8.4375, -5.25) \} \][/tex]
Given the options, the correct solution set did not appear. However, based on our calculations, none of the options matches exactly. The correct answer is [tex]\(\{ (8.4375, -5.25) \}\)[/tex]. Since this specific option is not listed:
- The answer [tex]\(\emptyset \text{ (inconsistent system) }\)[/tex] is incorrect as we found a solution.
- [tex]\((45 / 4,-3)\)[/tex] and [tex]\((-45 / 4, -3)\)[/tex] do not align with our calculated solution.
Thus, based on our finding, the system has a solution, but it is not listed correctly in the provided options. The correct solution, as we calculated, is [tex]\((8.4375, -5.25)\)[/tex], which corresponds to the key methods we used to solve these linear equations.
[tex]\[ \begin{array}{l} \frac{1}{5} x-\frac{1}{4} y=3 \\ \frac{2}{3} x+\frac{1}{2} y=3 \end{array} \][/tex]
Step-by-Step Solution:
1. Write the equations in a standard form:
[tex]\[ \frac{1}{5} x - \frac{1}{4} y = 3 \][/tex]
[tex]\[ \frac{2}{3} x + \frac{1}{2} y = 3 \][/tex]
2. Multiply both sides of each equation to get rid of the fractions:
For the first equation:
[tex]\[ 20 \left(\frac{1}{5} x - \frac{1}{4} y = 3 \right) \][/tex]
[tex]\[ 4x - 5y = 60 \][/tex]
For the second equation:
[tex]\[ 6 \left(\frac{2}{3} x + \frac{1}{2} y = 3 \right) \][/tex]
[tex]\[ 4x + 3y = 18 \][/tex]
3. Rewrite the system of equations:
[tex]\[ \begin{array}{l} 4x - 5y = 60 \\ 4x + 3y = 18 \end{array} \][/tex]
4. Subtract the second equation from the first to eliminate [tex]\( x \)[/tex]:
[tex]\[ (4x - 5y) - (4x + 3y) = 60 - 18 \][/tex]
[tex]\[ -8y = 42 \][/tex]
5. Solve for [tex]\( y \)[/tex]:
[tex]\[ y = -\frac{42}{8} \][/tex]
[tex]\[ y = -5.25 \][/tex]
6. Substitute [tex]\( y = -5.25 \)[/tex] back into one of the original equations to solve for [tex]\( x \)[/tex]:
Substituting into [tex]\( 4x - 5y = 60 \)[/tex]:
[tex]\[ 4x - 5(-5.25) = 60 \][/tex]
[tex]\[ 4x + 26.25 = 60 \][/tex]
[tex]\[ 4x = 60 - 26.25 \][/tex]
[tex]\[ 4x = 33.75 \][/tex]
[tex]\[ x = \frac{33.75}{4} \][/tex]
[tex]\[ x = 8.4375 \][/tex]
7. The solution set for the system of equations is:
[tex]\[ \{ (8.4375, -5.25) \} \][/tex]
Given the options, the correct solution set did not appear. However, based on our calculations, none of the options matches exactly. The correct answer is [tex]\(\{ (8.4375, -5.25) \}\)[/tex]. Since this specific option is not listed:
- The answer [tex]\(\emptyset \text{ (inconsistent system) }\)[/tex] is incorrect as we found a solution.
- [tex]\((45 / 4,-3)\)[/tex] and [tex]\((-45 / 4, -3)\)[/tex] do not align with our calculated solution.
Thus, based on our finding, the system has a solution, but it is not listed correctly in the provided options. The correct solution, as we calculated, is [tex]\((8.4375, -5.25)\)[/tex], which corresponds to the key methods we used to solve these linear equations.
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