To solve the given system of equations by the method of addition (also known as the elimination method), follow these steps to eliminate one of the variables. Let's consider the system of equations:
[tex]\[
\begin{array}{l}
2x - 4y = 5 \\
6x - 3y = 10
\end{array}
\][/tex]
## Step 1: Identify the coefficients of the variable to be eliminated
First, choose a variable to eliminate. For this solution, let’s choose to eliminate \( x \). To do this, we need the coefficients of \( x \) in both equations to be equal in magnitude.
## Step 2: Make the coefficients of \( x \) equal in magnitude
The coefficients of \(x\) in the given equations are \(2\) in the first equation and \(6\) in the second equation. To eliminate \( x \), we need to make these coefficients the same. We can do this by multiplying the first equation by 3:
[tex]\[
3 \cdot (2x - 4y) = 3 \cdot 5
\][/tex]
This gives us:
[tex]\[
6x - 12y = 15
\][/tex]
Now, the system of equations is:
[tex]\[
\begin{array}{l}
6x - 12y = 15 \\
6x - 3y = 10
\end{array}
\][/tex]
## Step 3: Make the coefficients opposites
Next, to eliminate \( x \), we want the coefficients to be opposite in sign. To do this, we can multiply the second equation by -2:
[tex]\[
-2 \cdot (6x - 3y) = -2 \cdot 10
\][/tex]
This gives us:
[tex]\[
-12x + 6y = -20
\][/tex]
Now our system of equations is:
[tex]\[
\begin{array}{l}
6x - 12y = 15 \\
-12x + 6y = -20
\end{array}
\][/tex]
So, the system after making the coefficients of \( x \) equal in magnitude but opposite in sign is:
[tex]\[
\begin{array}{l}
6x - 12y = 15 \\
-12x + 6y = -20
\end{array}
\][/tex]
Now, add these two equations to eliminate the variable \( x \).
## Step 4: Add the equations
[tex]\[
(6x - 12y) + (-12x + 6y) = 15 + (-20)
\][/tex]
This simplifies to:
[tex]\[
-6x - 6y = -5
\][/tex]
In this form, one variable, \( x \), has been successfully eliminated, and we can now solve for \( y \).
Thus, before adding the equations, you should:
1. Multiply the first equation by 3.
2. Multiply the second equation by -2.
This step-by-step procedure ensures that one variable is eliminated when the equations are added.
