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Use the substitution method to solve the system 2x+2y=6

, 3x+2y=9
.(1 poi

3 answers

To solve the system of equations using the substitution method, we will first solve one of the equations for one variable and then substitute that expression into the other equation.

The system of equations is:

  1. \( 2x + 2y = 6 \)
  2. \( 3x + 2y = 9 \)

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

  1. \( 2x + 2y = 6 \)

Divide the entire equation by 2:

\[ x + y = 3 \]

Now, solve for \( y \):

\[ y = 3 - x \]

Next, substitute \( y \) in the second equation \( 3x + 2y = 9 \):

\[ 3x + 2(3 - x) = 9 \]

Now, simplify and solve for \( x \):

\[ 3x + 6 - 2x = 9 \]

Combine like terms:

\[ x + 6 = 9 \]

Subtract 6 from both sides:

\[ x = 3 \]

Now that we have \( x = 3 \), we can substitute it back into the equation we found for \( y \):

\[ y = 3 - x = 3 - 3 = 0 \]

So, the solution to the system of equations is:

\[ \boxed{(3, 0)} \]

Let's double-check by substituting \( x = 3 \) and \( y = 0 \) back into the original equations:

For the first equation:

\[ 2(3) + 2(0) = 6 \quad \text{which simplifies to} \quad 6 = 6 \quad \text{(True)} \]

For the second equation:

\[ 3(3) + 2(0) = 9 \quad \text{which simplifies to} \quad 9 = 9 \quad \text{(True)} \]

Both equations are satisfied, confirming that our solution is correct.

Thus, the final answer is:

\[ \boxed{(3, 0)} \]

Use the substitution method to solve the system 9x−3y=12
, y+6=2x
.

Let's solve the given system of equations using the substitution method. The equations are:

  1. \( 9x - 3y = 12 \)
  2. \( y + 6 = 2x \)

First, we can solve the second equation for \( y \):

\[ y + 6 = 2x \]

Subtract 6 from both sides to isolate \( y \):

\[ y = 2x - 6 \]

Now that we have \( y \) expressed in terms of \( x \), we can substitute this expression for \( y \) into the first equation:

\[ 9x - 3(2x - 6) = 12 \]

Now, distribute the \( -3 \):

\[ 9x - 6x + 18 = 12 \]

Combine like terms:

\[ 3x + 18 = 12 \]

Next, subtract 18 from both sides to isolate the term with \( x \):

\[ 3x = 12 - 18 \]

\[ 3x = -6 \]

Now, divide by 3:

\[ x = -2 \]

Now that we have the value of \( x \), we can substitute \( x = -2 \) back into the equation we derived for \( y \):

\[ y = 2(-2) - 6 \]

\[ y = -4 - 6 \]

\[ y = -10 \]

Thus, the solution to the system of equations is:

\[ \boxed{(-2, -10)} \]

Verification:

To ensure the solution is correct, plug \( x = -2 \) and \( y = -10 \) back into both original equations.

  1. For the first equation \( 9x - 3y = 12 \):

\[ 9(-2) - 3(-10) = -18 + 30 = 12 \quad \text{(True)} \]

  1. For the second equation \( y + 6 = 2x \):

\[ -10 + 6 = 2(-2) \]

\[ -4 = -4 \quad \text{(True)} \]

Both equations are indeed satisfied, confirming that the solution is correct:

\[ \boxed{(-2, -10)} \]