

The elimination method is ideal for solving this system of equations. By which number must you multiply the second equation to eliminate the y-variable, and what is the solution for this system?

x + 3y = 42
2x − y = 14


To use the elimination method, we need to make the coefficients of the \( y \)-variable in both equations equal (but opposite in sign) so that we can eliminate it. 

Given the system:

1. \( x + 3y = 42 \)
2. \( 2x - y = 14 \)

### Step 1: Eliminate the \( y \)-Variable
To eliminate \( y \), we need to make the coefficients of \( y \) in both equations the same. The first equation has \( 3y \) and the second has \( -y \). To make the coefficients match, we can multiply the second equation by 3:

\[
3(2x - y) = 3(14)
\]

which simplifies to:

\[
6x - 3y = 42
\]

### Step 2: Add the Equations
Now, we have:

1. \( x + 3y = 42 \)
2. \( 6x - 3y = 42 \)

Add these two equations together:

\[
(x + 6x) + (3y - 3y) = 42 + 42
\]

\[
7x = 84
\]

\[
x = 12
\]

### Step 3: Substitute to Find \( y \)
Now that we know \( x = 12 \), we can substitute this value into one of the original equations. Using the first equation:

\[
12 + 3y = 42
\]

\[
3y = 42 - 12
\]

\[
3y = 30
\]

\[
y = 10
\]

### Solution:
The solution for this system is:

\[
x = 12, \quad y = 10
\]

So, you need to multiply the second equation by 3 to eliminate the \( y \)-variable. 

Would you like further details or have any other questions?

Here are 5 related questions:
1. How can we verify the solution obtained using substitution?
2. What if we chose to eliminate the \( x \)-variable instead; how would the solution process change?
3. Can the elimination method be used for any system of linear equations?
4. How does the elimination method differ from the substitution method?
5. What are some situations where the elimination method is more efficient than substitution?

**Tip:** Always double-check your solution by substituting the values back into the original equations.

The elimination method is ideal for solving this system of equations. By which number must you multiply the second equation to eliminate the y-variable, and what is the solution for this system?

x + 3y = 42
2x − y = 14

Learn how to solve the system of equations x + 3y = 42 and 2x - y = 14 using the elimination method. This problem covers key algebraic techniques, including elimination by equation multiplication. Ideal for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation