Solving Simultaneous Equations Using Substitution Quick Check

Question

3 of 53 of 5 Items
Question
Use the substitution method to solve the system 2.5x+y=−2, 3x+2y=0.(1 point)
Responses

infinite solutions
infinite solutions

(−2,3)
left parenthesis negative 2 comma 3 right parenthesis

(3,−2)
left parenthesis 3 comma negative 2 right parenthesis

no solution
no solution
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Answer 1

To solve the system of equations using substitution, we start with the following equations:

1. \( 2.5x + y = -2 \)
2. \( 3x + 2y = 0 \)

Step 1: Solve one equation for one variable

Let's solve the first equation for \( y \):

\[ y = -2 - 2.5x \]

Step 2: Substitute into the second equation

Now we'll substitute \( y \) into the second equation:

\[ 3x + 2(-2 - 2.5x) = 0 \]

Distributing the \( 2 \):

\[ 3x - 4 - 5x = 0 \]

Step 3: Combine like terms

Combine the \( x \) terms:

\[ -2x - 4 = 0 \]

Step 4: Solve for \( x \)

Add 4 to both sides:

\[ -2x = 4 \]

Divide by -2:

\[ x = -2 \]

Step 5: Substitute back to find \( y \)

Now use this value of \( x \) to find \( y \):

\[ y = -2 - 2.5(-2) \]

Calculating:

\[ y = -2 + 5 = 3 \]

Step 6: Write the solution

The solution to the system of equations is:

\[ (x, y) = (-2, 3) \]

Thus, the correct response from the options given is:

(−2, 3).