Question
Solve the system of equations by any method. \[ \begin{array}{c}-2 y=-1 \\ 7 x-14 y=8\end{array} \] \[ \begin{array}{l}x+2 y\end{array} \]
Ask by Bolton Osborne. in the United States
Feb 27,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The solution to the system of equations is \( x = \frac{15}{7} \) and \( y = \frac{1}{2} \).
Solution
Solve the system of equations \( -2y=-1;7x-14y=8 \).
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
\(\left\{ \begin{array}{l}-2y=-1\\7x-14y=8\end{array}\right.\)
- step1: Solve the equation:
\(\left\{ \begin{array}{l}y=\frac{1}{2}\\7x-14y=8\end{array}\right.\)
- step2: Substitute the value of \(y:\)
\(7x-14\times \frac{1}{2}=8\)
- step3: Multiply the numbers:
\(7x-7=8\)
- step4: Move the constant to the right side:
\(7x=8+7\)
- step5: Add the numbers:
\(7x=15\)
- step6: Divide both sides:
\(\frac{7x}{7}=\frac{15}{7}\)
- step7: Divide the numbers:
\(x=\frac{15}{7}\)
- step8: Calculate:
\(\left\{ \begin{array}{l}x=\frac{15}{7}\\y=\frac{1}{2}\end{array}\right.\)
- step9: Check the solution:
\(\left\{ \begin{array}{l}x=\frac{15}{7}\\y=\frac{1}{2}\end{array}\right.\)
- step10: Rewrite:
\(\left(x,y\right) = \left(\frac{15}{7},\frac{1}{2}\right)\)
The solution to the system of equations is \( (x, y) = \left(\frac{15}{7}, \frac{1}{2}\right) \) or \( (x, y) = (2.142857, 0.5) \).
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Simplify this solution Mind Expander
To solve the system of equations, we start with the first equation: \[ -2y = -1 \] Dividing both sides by -2 gives: \[ y = \frac{1}{2} \] Now, substitute \(y\) into the second equation: \[ 7x - 14y = 8 \] Replacing \(y\) yields: \[ 7x - 14\left(\frac{1}{2}\right) = 8 \] This simplifies to: \[ 7x - 7 = 8 \] Adding 7 to both sides results in: \[ 7x = 15 \] Finally, dividing by 7, we find: \[ x = \frac{15}{7} \] Now that we have \(x\) and \(y\), we can evaluate \(x + 2y\): \[ x + 2y = \frac{15}{7} + 2\left(\frac{1}{2}\right) = \frac{15}{7} + 1 = \frac{15}{7} + \frac{7}{7} = \frac{22}{7} \] So, the final answer is: \[ x + 2y = \frac{22}{7} \]
