To solve this system of equations graphically, we will go through the following steps:
1. Rewrite and understand each equation.
2. Plot each line on the graph.
3. Find the intersection point of the two lines, which represents the solution to the system.
### Step 1: Rewrite and Understand Each Equation
#### Equation 1: [tex]\( y = -x + 6 \)[/tex]
This is already in slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] (the slope) is [tex]\(-1\)[/tex] and [tex]\( b \)[/tex] (the y-intercept) is [tex]\( 6 \)[/tex]. This means the line crosses the y-axis at [tex]\( (0, 6) \)[/tex] and has a downward slope of [tex]\(-1\)[/tex].
#### Equation 2: [tex]\( 5x - 2y = 16 \)[/tex]
First, we need to solve for [tex]\( y \)[/tex]:
[tex]\[ 5x - 2y = 16 \][/tex]
[tex]\[ -2y = -5x + 16 \][/tex]
[tex]\[ y = \frac{5}{2}x - 8 \][/tex]
This line is also in slope-intercept form, where [tex]\( m = \frac{5}{2} \)[/tex] (the slope) and [tex]\( b = -8 \)[/tex] (the y-intercept). This line crosses the y-axis at [tex]\( (0, -8) \)[/tex] and has an upward slope of [tex]\( \frac{5}{2} \)[/tex] or [tex]\( 2.5 \)[/tex].
### Step 2: Plot Each Line on the Graph
We'll use the significant points we have:
- For [tex]\( y = -x + 6 \)[/tex]:
- y-intercept: [tex]\( (0, 6) \)[/tex]
- Another point using x = 6: [tex]\( y = -6 + 6 = 0 \)[/tex], so [tex]\( (6, 0) \)[/tex]
- For [tex]\( y = \frac{5}{2}x - 8 \)[/tex]:
- y-intercept: [tex]\( (0, -8) \)[/tex]
- Another point using x = 4: [tex]\( y = \frac{5}{2}(4) - 8 = 10 - 8 = 2 \)[/tex], so [tex]\( (4, 2) \)[/tex]
### Step 3: Find the Intersection Point
We find the intersection of the two lines to determine the solution. Set the equations equal:
[tex]\[ -x + 6 = \frac{5}{2}x - 8 \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ 6 + 8 = \frac{5}{2}x + x \][/tex]
[tex]\[ 14 = \frac{7}{2} x \][/tex]
[tex]\[ x = \frac{14 \times 2}{7} \][/tex]
[tex]\[ x = 4 \][/tex]
Now substitute [tex]\( x = 4 \)[/tex] back into either equation to find [tex]\( y \)[/tex]:
Using [tex]\( y = -x + 6 \)[/tex]:
[tex]\[ y = -4 + 6 \][/tex]
[tex]\[ y = 2 \][/tex]
So, the intersection point, and hence the solution, is [tex]\( (4, 2) \)[/tex].
### Graphical Representation:
- The first line [tex]\( y = -x + 6 \)[/tex] passes through the points [tex]\( (0, 6) \)[/tex] and [tex]\( (6, 0) \)[/tex].
- The second line [tex]\( y = \frac{5}{2}x - 8 \)[/tex] passes through the points [tex]\( (0, -8) \)[/tex] and [tex]\( (4, 2) \)[/tex].
The intersection point of these two lines is [tex]\((4, 2)\)[/tex], which is the solution to the system of equations.
This completes the graphical solution of the given system.
