To solve the system of equations given by:
- \( 2x + 3y = 13 \)
- \( x = 2 \)
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 \( 4 + 3y = 13 \):
-
Subtract \( 4 \) from both sides:
\[ 3y = 13 - 4 \]
\[ 3y = 9 \] -
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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