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Solve using the substitution method. \( \begin{array}{l}3 x-y=5 \\ 5 y=15 x-25\end{array} \) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. There is only one solution. The solution of the system is (Simplify your answer. Type an ordered pair.) B. There are infinitely many solutions of the form ( x . (Simplify your answer.) C. There is no solution.

Ask by Reyes Pope. in the United States
Feb 22,2025

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There are infinitely many solutions of the form (x, 3x – 5).

Solución

We are given the system:   3x – y = 5    (1)   5y = 15x – 25  (2) Step 1. Solve equation (1) for y:   3x – y = 5 ⟹ y = 3x – 5 Step 2. Substitute y = 3x – 5 into equation (2):   5(3x – 5) = 15x – 25   15x – 25 = 15x – 25 Since both sides are identical, this equation holds true for all values of x. This shows that the system does not restrict x further and y is given by y = 3x – 5. Thus, the system has infinitely many solutions of the form (x, 3x – 5). The correct choice is: B. There are infinitely many solutions of the form (x, 3x – 5).

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The Deep Dive

Let's start by solving the system of equations step by step using the substitution method. 1. From the first equation, we can express \(y\) in terms of \(x\): \[ 3x - y = 5 \implies y = 3x - 5 \] 2. Now, we can substitute this expression for \(y\) into the second equation: \[ 5y = 15x - 25 \implies 5(3x - 5) = 15x - 25 \] Simplifying this: \[ 15x - 25 = 15x - 25 \] 3. This simplifies to a statement that is always true (15x - 25 = 15x - 25), which means that there are infinitely many solutions. Now choosing the correct option: B. There are infinitely many solutions of the form ( x , 3x - 5 ).

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