Find the best answers to your questions at Westonci.ca, where experts and enthusiasts provide accurate, reliable information. Connect with a community of experts ready to provide precise solutions to your questions quickly and accurately. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
To determine the value of \( Q \) for which the given system of equations
[tex]\[ \begin{cases} x - 3y = 4 \\ Qx - 6y = 8 \end{cases} \][/tex]
has the solution \(\{(x, y) : x - 3y = 4\}\), follow these steps:
1. Let's start with examining the first equation:
[tex]\[ x - 3y = 4 \][/tex]
Notice that this equation is already in a simplified form.
2. Now, consider the second equation:
[tex]\[ Qx - 6y = 8 \][/tex]
3. We need to rewrite the second equation in a form that is comparable to the first equation. Notice that \(Qx - 6y = 8\) can be factored out to show a relation comparable to the first equation. Let's divide the second equation by 2 to simplify it:
[tex]\[ \frac{Qx - 6y}{2} = \frac{8}{2} \][/tex]
Simplifying both sides we get:
[tex]\[ \frac{Q}{2} x - 3y = 4 \][/tex]
4. Now observe that for this simplified version of the second equation to be consistent with the first equation \(x - 3y = 4\), the coefficients of \(x\) and \(y\) on the left side must match between the two equations.
5. By comparing the coefficients of \(x\), we have:
[tex]\[ \frac{Q}{2} x = x \][/tex]
For this equality to hold true for all \(x\), it must be that:
[tex]\[ \frac{Q}{2} = 1 \][/tex]
6. Solving for \(Q\):
[tex]\[ Q = 2 \times 1 \][/tex]
[tex]\[ Q = 2 \][/tex]
Thus, the value of [tex]\(Q\)[/tex] that ensures the system of equations has the consistent solution [tex]\(\{(x, y) : x - 3y = 4\}\)[/tex] is [tex]\(\boxed{2}\)[/tex].
[tex]\[ \begin{cases} x - 3y = 4 \\ Qx - 6y = 8 \end{cases} \][/tex]
has the solution \(\{(x, y) : x - 3y = 4\}\), follow these steps:
1. Let's start with examining the first equation:
[tex]\[ x - 3y = 4 \][/tex]
Notice that this equation is already in a simplified form.
2. Now, consider the second equation:
[tex]\[ Qx - 6y = 8 \][/tex]
3. We need to rewrite the second equation in a form that is comparable to the first equation. Notice that \(Qx - 6y = 8\) can be factored out to show a relation comparable to the first equation. Let's divide the second equation by 2 to simplify it:
[tex]\[ \frac{Qx - 6y}{2} = \frac{8}{2} \][/tex]
Simplifying both sides we get:
[tex]\[ \frac{Q}{2} x - 3y = 4 \][/tex]
4. Now observe that for this simplified version of the second equation to be consistent with the first equation \(x - 3y = 4\), the coefficients of \(x\) and \(y\) on the left side must match between the two equations.
5. By comparing the coefficients of \(x\), we have:
[tex]\[ \frac{Q}{2} x = x \][/tex]
For this equality to hold true for all \(x\), it must be that:
[tex]\[ \frac{Q}{2} = 1 \][/tex]
6. Solving for \(Q\):
[tex]\[ Q = 2 \times 1 \][/tex]
[tex]\[ Q = 2 \][/tex]
Thus, the value of [tex]\(Q\)[/tex] that ensures the system of equations has the consistent solution [tex]\(\{(x, y) : x - 3y = 4\}\)[/tex] is [tex]\(\boxed{2}\)[/tex].
Thanks for using our platform. We're always here to provide accurate and up-to-date answers to all your queries. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.