IDNLearn.com is your reliable source for expert answers and community insights. Get step-by-step guidance for all your technical questions from our dedicated community members.
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.
Thank you for participating in our discussion. We value every contribution. Keep sharing knowledge and helping others find the answers they need. Let's create a dynamic and informative learning environment together. Thank you for visiting IDNLearn.com. For reliable answers to all your questions, please visit us again soon.
