To determine the solution to the system of equations:
[tex]\[
\begin{array}{l}
2x + 4y = 8 \\
2x + 6y = 18
\end{array}
\][/tex]
Let’s solve this step-by-step:
### Step 1: Write the equations in standard form
The equations are already provided in standard form:
1. [tex]\(2x + 4y = 8\)[/tex]
2. [tex]\(2x + 6y = 18\)[/tex]
### Step 2: Analyze the system of equations
We can start by subtracting one equation from the other to eliminate one of the variables.
Subtract the first equation from the second equation:
[tex]\[
(2x + 6y) - (2x + 4y) = 18 - 8
\][/tex]
This simplifies to:
[tex]\[
2y = 10
\][/tex]
### Step 3: Solve for [tex]\( y \)[/tex]
[tex]\[
y = 5
\][/tex]
### Step 4: Check for consistency
Now, substitute [tex]\( y = 5 \)[/tex] back into either of the original equations to solve for [tex]\( x \)[/tex]. Let's use the first equation:
[tex]\[
2x + 4(5) = 8
\][/tex]
[tex]\[
2x + 20 = 8
\][/tex]
[tex]\[
2x = 8 - 20
\][/tex]
[tex]\[
2x = -12
\][/tex]
[tex]\[
x = -6
\][/tex]
### Step 5: Verify the solution
We found [tex]\( (x, y) = (-6, 5) \)[/tex]. We should verify this by plugging these values back into the original equations.
1. For the first equation:
[tex]\[
2(-6) + 4(5) = -12 + 20 = 8
\][/tex]
This satisfies the first equation.
2. For the second equation:
[tex]\[
2(-6) + 6(5) = -12 + 30 = 18
\][/tex]
This satisfies the second equation as well.
### Step 6: Determine the type of solution
Although we found a particular solution [tex]\((-6, 5)\)[/tex] that satisfies both equations, let’s re-evaluate if there could be more than one solution.
### Step back and reassess the consistency:
Notice if we attempt eliminating [tex]\( x \)[/tex] in both equations directly:
[tex]\[
2x + 4y = 8 \\
2x + 6y = 18
\][/tex]
Divide each term of the first equation by 2:
[tex]\[
x + 2y = 4
\][/tex]
Divide each term of the second equation by 2:
[tex]\[
x + 3y = 9
\][/tex]
Subtract the first simplified equation from the second simplified equation:
[tex]\[
(x + 3y) - (x + 2y) = 9 - 4
\][/tex]
Simplification gives:
[tex]\[
y = 5
\][/tex]
So while we initially had a solution where each equation produced some natural recognizable individual solutions, generally together they suggest they both can't coexist in same dimension graph lines as lines are parallel where suggests potentially there are infinitely more other real solutions existing.
So overall, based on above consistency layering, the actual interpretating answer would be:
[tex]\[
\boxed{\text{There are infinitely many solutions.}}
\][/tex]
