Question
What is the solution to the system of equations? \( \left\{\begin{array}{l}4 x-5 y=-20 \\ -4 x+5 y=20\end{array}\right. \)
Ask by Black Hill. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The system of equations has infinitely many solutions.
Solution
Solve the system of equations \( 4x-5y=-20;-4x+5y=20 \).
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
\(\left\{ \begin{array}{l}4x-5y=-20\\-4x+5y=20\end{array}\right.\)
- step1: Solve the equation:
\(\left\{ \begin{array}{l}x=\frac{-20+5y}{4}\\-4x+5y=20\end{array}\right.\)
- step2: Substitute the value of \(x:\)
\(-4\times \frac{-20+5y}{4}+5y=20\)
- step3: Simplify:
\(20=20\)
- step4: Calculate:
\(\left(x,y\right) = \left(\frac{-20+5y}{4},y\right),y \in \mathbb{R}\)
- step5: Alternative Form:
\(\textrm{Infinitely many solutions}\)
The system of equations has infinitely many solutions.
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
To solve this system of equations, let's add both equations together. When we do this: \( (4x - 5y) + (-4x + 5y) = -20 + 20 \) The \(4x\) and \(-4x\) cancel out, and \(-5y + 5y\) also cancels, leading us to: \( 0 = 0 \) This means that the two equations represent the same line, thus there are infinitely many solutions along that line. To express the relationship between \(x\) and \(y\), we can solve one of the original equations for \(y\). From the first equation: \( 4x - 5y = -20 \) can be rearranged to \( 5y = 4x + 20 \) giving \( y = \frac{4}{5}x + 4 \). So, all points \((x, y)\) that satisfy this equation are solutions to the system!
