Answer the questions and complete the steps to solve the system of equations below.

2x + 3y = 13
x = 2
a) Which variable should you substitute for in the equation 2x + 3y = 13?
choose your answer...
b) What value can be substituted for that variable? choose your answer...
c) Rewrite the equation with the substitution. choose your answer...
d) Solve the equation for y.
e) Use substitution to check your solution for the system.

1 answer

To solve the system of equations given by:

  1. \( 2x + 3y = 13 \)
  2. \( x = 2 \)

let's go through each step:

a) Which variable should you substitute for in the equation \( 2x + 3y = 13 \)?
The variable to substitute for is \( x \) (because we have a direct value for \( x\)).

b) What value can be substituted for that variable?
The value that can be substituted for \( x \) is \( 2 \) (from the equation \( x = 2 \)).

c) Rewrite the equation with the substitution.
Substituting \( x = 2 \) into the equation \( 2x + 3y = 13 \) gives:
\[ 2(2) + 3y = 13 \]
This simplifies to:
\[ 4 + 3y = 13 \]

d) Solve the equation for \( y \).
Now, we solve \( 4 + 3y = 13 \):

  1. Subtract \( 4 \) from both sides:
    \[ 3y = 13 - 4 \]
    \[ 3y = 9 \]

  2. Divide both sides by \( 3 \):
    \[ y = 3 \]

So, the solution for \( y \) is \( 3 \).

e) Use substitution to check your solution for the system.
Now, we need to check the solution \( (x, y) = (2, 3) \) in both equations:

For \( 2x + 3y = 13 \):
\[ 2(2) + 3(3) = 4 + 9 = 13 \]
This equation is satisfied.

For \( x = 2 \):
This is already satisfied, as it is given.

Thus, the solution to the system of equations is \( x = 2 \) and \( y = 3 \).

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