Question
Solve by the substitution method. (If there is no solution, enter NO SOLUTION. Use the parameters \( x \) and \( y \) as necessary.) \[ \begin{aligned} 2 x-3 y=4 \\ x=5 y+2\end{aligned} \] \( (x, y)=( \)
Ask by Gardner Cook. in the United States
Dec 04,2024
Real Tutor Solution
Tutor-Verified Answer
Answer
\( (x, y) = (2, 0) \).
Solution
Solve the system of equations \( 2x-3y=4;x=5y+2 \).
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
\(\left\{ \begin{array}{l}2x-3y=4\\x=5y+2\end{array}\right.\)
- step1: Substitute the value of \(x:\)
\(2\left(5y+2\right)-3y=4\)
- step2: Simplify:
\(7y+4=4\)
- step3: Move the constant to the right side:
\(7y=4-4\)
- step4: Subtract the terms:
\(7y=0\)
- step5: Rewrite the expression:
\(y=0\)
- step6: Substitute the value of \(y:\)
\(x=5\times 0+2\)
- step7: Simplify:
\(x=2\)
- step8: Calculate:
\(\left\{ \begin{array}{l}x=2\\y=0\end{array}\right.\)
- step9: Check the solution:
\(\left\{ \begin{array}{l}x=2\\y=0\end{array}\right.\)
- step10: Rewrite:
\(\left(x,y\right) = \left(2,0\right)\)
The solution to the system of equations by the substitution method is \( (x, y) = (2, 0) \).
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution The Deep Dive
To solve the system of equations using the substitution method, we start with the second equation: 1. From the equation \( x = 5y + 2 \), we can substitute \( x \) into the first equation \( 2x - 3y = 4 \): \[ 2(5y + 2) - 3y = 4 \] 2. Now expand and simplify: \[ 10y + 4 - 3y = 4 \] \[ 10y - 3y + 4 = 4 \] \[ 7y + 4 = 4 \] 3. Next, isolate \( y \): \[ 7y = 4 - 4 \] \[ 7y = 0 \] \[ y = 0 \] 4. Now substitute \( y = 0 \) back into the expression for \( x \): \[ x = 5(0) + 2 = 2 \] Thus, the solution to the system is: \( (x, y) = (2, 0) \)
