Answer :
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.
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.
