To solve the system of equations graphically and determine the solution, let's first graph each equation and then analyze their intersection.
The given system of equations is:
1. [tex]\(4x + 12y = 12\)[/tex]
2. [tex]\(2x + 6y = 12\)[/tex]
### Step 1: Rewrite the equations in slope-intercept form ([tex]\(y = mx + b\)[/tex])
#### Equation 1: [tex]\(4x + 12y = 12\)[/tex]
Let's solve for [tex]\(y\)[/tex]:
[tex]\[ 4x + 12y = 12 \][/tex]
Subtract [tex]\(4x\)[/tex] from both sides:
[tex]\[ 12y = 12 - 4x \][/tex]
Divide by 12:
[tex]\[ y = 1 - \frac{x}{3} \][/tex]
So, the slope-intercept form of the first equation is:
[tex]\[ y = -\frac{1}{3}x + 1 \][/tex]
#### Equation 2: [tex]\(2x + 6y = 12\)[/tex]
Similarly, solving for [tex]\(y\)[/tex]:
[tex]\[ 2x + 6y = 12 \][/tex]
Subtract [tex]\(2x\)[/tex] from both sides:
[tex]\[ 6y = 12 - 2x \][/tex]
Divide by 6:
[tex]\[ y = 2 - \frac{x}{3} \][/tex]
So, the slope-intercept form of the second equation is:
[tex]\[ y = -\frac{1}{3}x + 2 \][/tex]
### Step 2: Graph the equations
1. Graph of [tex]\( y = -\frac{1}{3}x + 1 \)[/tex]
- The y-intercept is 1 (point (0, 1)).
- The slope is [tex]\(-\frac{1}{3}\)[/tex], which means for every 1 unit increase in [tex]\(x\)[/tex], [tex]\(y\)[/tex] decreases by [tex]\(\frac{1}{3}\)[/tex].
2. Graph of [tex]\( y = -\frac{1}{3}x + 2 \)[/tex]
- The y-intercept is 2 (point (0, 2)).
- The slope is [tex]\(-\frac{1}{3}\)[/tex], same as the first equation.
### Step 3: Analyze the graphs
Both equations have the same slope [tex]\(-\frac{1}{3}\)[/tex], meaning the lines are parallel. However, they have different y-intercepts (1 and 2).
### Conclusion: Determine the type of solution
Because the lines are parallel and have different y-intercepts, they do not intersect at any point. Therefore, there is no solution to the system of equations.
Thus, the system of equations has no solution.
The final answer is:
- There is no solution.
