6. Use the substitution method to solve the system

Question

x-2y=3 & 4x-8y=12
Show your work

Answer 1

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.

Given the system of equations:

1. \( x - 2y = 3 \) (Equation 1)
2. \( 4x - 8y = 12 \) (Equation 2)

Step 1: Solve Equation 1 for x

From Equation 1, we can solve for \( x \):

\[ x = 2y + 3 \]

Step 2: Substitute into Equation 2

Now, we substitute \( x \) from Equation 1 into Equation 2:

\[ 4(2y + 3) - 8y = 12 \]

Step 3: Simplify the equation

Now, expand and simplify:

\[ 8y + 12 - 8y = 12 \]

Notice that \( 8y - 8y \) cancels out:

\[ 12 = 12 \]

Step 4: Interpretation

The statement \( 12 = 12 \) is always true, which means that the original system of equations represents the same line. Thus, there are infinitely many solutions along that line.

Step 5: Expressing the solution

We can express the solution set in terms of one parameter. We already have:

\[ x = 2y + 3 \]

We can let \( y = t \), where \( t \) represents any real number.

Then we can express \( x \) as:

\[ x = 2t + 3 \]

Final Result

The solution to the system can be expressed in parametric form as:

\[ (x, y) = (2t + 3, t) \quad \text{for any real number } t. \]

This means that there are infinitely many solutions, all lying on the line defined by the original equations.

Answer 2

Shorter

Answer 3

Sure! Here’s a more concise version of the solution using substitution:

Given the system of equations:

1. \( x - 2y = 3 \)
2. \( 4x - 8y = 12 \)

Step 1: Solve Equation 1 for x

From Equation 1:

\[ x = 2y + 3 \]

Step 2: Substitute into Equation 2

Substitute \( x \) into Equation 2:

\[ 4(2y + 3) - 8y = 12 \]

Step 3: Simplify

Expand and simplify:

\[ 8y + 12 - 8y = 12 \]

This simplifies to:

\[ 12 = 12 \]

Conclusion

This true statement indicates that the equations are equivalent, meaning there are infinitely many solutions.

The general solution can be expressed as:

\[ (x, y) = (2t + 3, t) \quad \text{for any real number } t. \]

Answer 4

More shorter please

Answer 5

Sure! Here’s an even more concise version:

Given the system:

1. \( x - 2y = 3 \)
2. \( 4x - 8y = 12 \)

Steps:

1. Solve \( x - 2y = 3 \) for \( x \):
   \[ x = 2y + 3 \]

2. Substitute into the second equation:
   \[ 4(2y + 3) - 8y = 12 \]
   \[ 12 = 12 \] (Always true)

Conclusion:

Infinitely many solutions. Expressed as:
\[ (x, y) = (2t + 3, t) \quad \text{for any real number } t. \]