Question
below have a solution? Explair \[ \left\{\begin{array}{l}y=\frac{5}{3} x+15 \\ 5 x-3 y=6\end{array}\right. \]
Ask by Bartlett Tran. in the United States
Feb 26,2025
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Tutor-Verified Answer
Answer
The system of equations has no solution.
Solution
Solve the system of equations \( \begin{cases} y=\frac{5}{3}x+15 \\ 5x-3y=6 \end{cases} \).
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
\(\left\{ \begin{array}{l}y=\frac{5}{3}x+15\\5x-3y=6\end{array}\right.\)
- step1: Substitute the value of \(y:\)
\(5x-3\left(\frac{5}{3}x+15\right)=6\)
- step2: Simplify:
\(-45=6\)
- step3: Calculate:
\(\left\{ \begin{array}{l}x \in \varnothing \\y \in \varnothing \end{array}\right.\)
- step4: Rewrite:
\((x, y) \in \varnothing\)
The system of equations has no solution. This means that there is no value of \( x \) and \( y \) that satisfies both equations simultaneously.
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Simplify this solution Extra Insights
To determine if the system of equations has a solution, we can either graph both equations or solve them algebraically. Here, let’s do the algebraic method! Starting with the first equation, \( y = \frac{5}{3} x + 15 \), we can substitute this expression for \( y \) into the second equation: \[ 5x - 3\left(\frac{5}{3} x + 15\right) = 6 \] Now, simplifying this gives: \[ 5x - 5x - 45 = 6 \] This leads to: \[ -45 = 6 \] Since this statement is false, it indicates that the two lines are parallel and do not intersect. Thus, the system of equations has no solution!
