To solve the system of equations
[tex]\[ 4.2x + 8y = 41.8 \][/tex]
[tex]\[ -4.2x + y = 19.4 \][/tex]
using the linear combination (or elimination) method, we will follow these steps:
### Step 1: Combine the Equations
First, observe that the coefficients of [tex]\( x \)[/tex] in the two equations are opposites of each other, i.e., [tex]\( 4.2 \)[/tex] and [tex]\( -4.2 \)[/tex]. This makes it ideal for elimination.
Add the two equations together to eliminate [tex]\( x \)[/tex]:
[tex]\[ (4.2x + 8y) + (-4.2x + y) = 41.8 + 19.4 \][/tex]
### Step 2: Simplify the Resulting Equation
When we add them, the [tex]\( x \)[/tex] terms cancel each other out:
[tex]\[ 4.2x - 4.2x + 8y + y = 41.8 + 19.4 \][/tex]
[tex]\[ 0 + 9y = 61.2 \][/tex]
[tex]\[ 9y = 61.2 \][/tex]
### Step 3: Solve for [tex]\( y \)[/tex]
Divide both sides of the equation by 9 to solve for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{61.2}{9} \][/tex]
[tex]\[ y ≈ 6.8 \][/tex]
### Step 4: Substitute [tex]\( y \)[/tex] back into one of the original equations
Next, we substitute [tex]\( y ≈ 6.8 \)[/tex] into one of the original equations to solve for [tex]\( x \)[/tex]. Let's use the second equation:
[tex]\[ -4.2x + y = 19.4 \][/tex]
[tex]\[ -4.2x + 6.8 = 19.4 \][/tex]
### Step 5: Solve for [tex]\( x \)[/tex]
Isolate the [tex]\( x \)[/tex] term and solve for [tex]\( x \)[/tex]:
[tex]\[ -4.2x = 19.4 - 6.8 \][/tex]
[tex]\[ -4.2x = 12.6 \][/tex]
[tex]\[ x = \frac{12.6}{-4.2} \][/tex]
[tex]\[ x = -3 \][/tex]
So, the value of [tex]\( x \)[/tex] is [tex]\( -3 \)[/tex].
