To graph the equation [tex]\(6x + 2y = 12\)[/tex] using the intercepts, we will follow a detailed step-by-step approach to find the x-intercept and y-intercept and then use these points to draw the line on the graph.
### Step 1: Find the X-Intercept
The x-intercept is the point where the line crosses the x-axis. At this point, the value of [tex]\(y\)[/tex] is zero.
Set [tex]\(y = 0\)[/tex] in the equation:
[tex]\[6x + 2(0) = 12\][/tex]
[tex]\[6x = 12\][/tex]
[tex]\[x = \frac{12}{6}\][/tex]
[tex]\[x = 2\][/tex]
So, the x-intercept is [tex]\((2, 0)\)[/tex].
### Step 2: Find the Y-Intercept
The y-intercept is the point where the line crosses the y-axis. At this point, the value of [tex]\(x\)[/tex] is zero.
Set [tex]\(x = 0\)[/tex] in the equation:
[tex]\[6(0) + 2y = 12\][/tex]
[tex]\[2y = 12\][/tex]
[tex]\[y = \frac{12}{2}\][/tex]
[tex]\[y = 6\][/tex]
So, the y-intercept is [tex]\((0, 6)\)[/tex].
### Step 3: Plot the Intercepts
Now, we need to plot the intercepts on the coordinate plane.
- Plot the x-intercept [tex]\((2, 0)\)[/tex]. This point is 2 units to the right of the origin (0,0) on the x-axis.
- Plot the y-intercept [tex]\((0, 6)\)[/tex]. This point is 6 units above the origin (0,0) on the y-axis.
### Step 4: Draw the Line
Using a straight edge, draw a line passing through the two intercepts [tex]\((2, 0)\)[/tex] and [tex]\((0, 6)\)[/tex]. This line represents the equation [tex]\(6x + 2y = 12\)[/tex].
The equation has been graphed correctly utilizing its intercepts, confirming the placement and trajectory of the line.
### Visual Representation
You would typically use graphing paper or a graphing tool for the accurate plotting, but if we describe the line visually:
- Begin at the y-intercept at [tex]\((0, 6)\)[/tex].
- Draw a straight line downwards crossing the point [tex]\((2, 0)\)[/tex] on the x-axis.
The drawn line accurately represents the equation [tex]\(6x + 2y = 12\)[/tex].
