practice synthetic division2.pdf - Name: Period: Date: Practice ... - Free Printable

Problem Analysis:

The task involves solving a system of linear equations using the substitution method. The given system of equations is:

1. \( y = 2x - 3 \)
2. \( x + y = 5 \)

We are asked to solve for \( x \) and \( y \), and then verify our solution by substituting back into both equations.

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Step-by-Step Solution:

#### Step 1: Understand the substitution method
The substitution method involves solving one equation for one variable (in this case, \( y \) is already solved in terms of \( x \) in the first equation) and substituting that expression into the other equation to solve for the remaining variable.

#### Step 2: Substitute \( y = 2x - 3 \) into the second equation
The second equation is:
\[ x + y = 5 \]

Substitute \( y = 2x - 3 \) into this equation:
\[ x + (2x - 3) = 5 \]

#### Step 3: Simplify and solve for \( x \)
Combine like terms:
\[ x + 2x - 3 = 5 \]
\[ 3x - 3 = 5 \]

Add 3 to both sides:
\[ 3x = 8 \]

Divide by 3:
\[ x = \frac{8}{3} \]

#### Step 4: Solve for \( y \) using \( y = 2x - 3 \)
Now that we have \( x = \frac{8}{3} \), substitute this value into the equation \( y = 2x - 3 \):
\[ y = 2\left(\frac{8}{3}\right) - 3 \]

Simplify:
\[ y = \frac{16}{3} - 3 \]

Convert 3 to a fraction with a denominator of 3:
\[ y = \frac{16}{3} - \frac{9}{3} \]
\[ y = \frac{7}{3} \]

#### Step 5: Write the solution as an ordered pair
The solution to the system of equations is:
\[ \left( x, y \right) = \left( \frac{8}{3}, \frac{7}{3} \right) \]

#### Step 6: Verify the solution
To verify, substitute \( x = \frac{8}{3} \) and \( y = \frac{7}{3} \) back into both original equations.

##### Verify in the first equation: \( y = 2x - 3 \)
\[ y = 2\left(\frac{8}{3}\right) - 3 \]
\[ y = \frac{16}{3} - 3 \]
\[ y = \frac{16}{3} - \frac{9}{3} \]
\[ y = \frac{7}{3} \]

This is correct.

##### Verify in the second equation: \( x + y = 5 \)
\[ x + y = \frac{8}{3} + \frac{7}{3} \]
\[ x + y = \frac{15}{3} \]
\[ x + y = 5 \]

This is also correct.

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Final Answer:

The solution to the system of equations is:
\[
\boxed{\left( \frac{8}{3}, \frac{7}{3} \right)}
\]