Answer :
Alright, let's go through the step-by-step process for graphing this system of linear equations:
### Step 1: Understand the System of Equations
We have two equations:
1. [tex]\( 1.15x + 0.65y = 8.90 \)[/tex]
2. [tex]\( x - 3y = -15 \)[/tex]
### Step 2: Find the Intersection Point
The solution to this system is the point where the two lines intersect. Given the solution:
[tex]\[ x = 4.13414634146341 \][/tex]
[tex]\[ y = 6.37804878048780 \][/tex]
So, the intersection point is approximately [tex]\((4.13, 6.38)\)[/tex].
### Step 3: Rewrite Each Equation in Slope-Intercept Form
To graph the lines, it's often easiest to rewrite the equations in the form [tex]\( y = mx + b \)[/tex] where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
#### For the first equation:
[tex]\[ 1.15x + 0.65y = 8.90 \][/tex]
To solve for [tex]\( y \)[/tex]:
[tex]\[ 0.65y = -1.15x + 8.90 \][/tex]
[tex]\[ y = \frac{-1.15}{0.65}x + \frac{8.90}{0.65} \][/tex]
[tex]\[ y = -1.76923x + 13.69231 \][/tex]
So, the first equation in slope-intercept form is:
[tex]\[ y = -1.76923x + 13.69231 \][/tex]
#### For the second equation:
[tex]\[ x - 3y = -15 \][/tex]
To solve for [tex]\( y \)[/tex]:
[tex]\[ -3y = -x - 15 \][/tex]
[tex]\[ y = \frac{1}{3}x + 5 \][/tex]
So, the second equation in slope-intercept form is:
[tex]\[ y = \frac{1}{3}x + 5 \][/tex]
### Step 4: Plot Each Line on the Graph
#### First Equation: [tex]\( y = -1.76923x + 13.69231 \)[/tex]
- The y-intercept ([tex]\(b\)[/tex]) is approximately 13.69, so the line crosses the y-axis at [tex]\((0, 13.69)\)[/tex].
- The slope ([tex]\(m\)[/tex]) is approximately [tex]\(-1.77\)[/tex].
- From the y-intercept, for every unit we move to the right on the x-axis, the line falls by about 1.77 units.
#### Second Equation: [tex]\( y = \frac{1}{3}x + 5 \)[/tex]
- The y-intercept ([tex]\(b\)[/tex]) is 5, so the line crosses the y-axis at [tex]\((0, 5)\)[/tex].
- The slope ([tex]\(\frac{1}{3}\)[/tex]) means that for every unit we move to the right on the x-axis, the line rises by 1/3 of a unit.
### Step 5: Graph the Lines
1. First Line:
- Start at [tex]\((0, 13.69)\)[/tex].
- Use the slope [tex]\(-1.77\)[/tex] to find another point. For example, move 1 unit to the right (to [tex]\(x=1\)[/tex]), and then move down 1.77 units (to [tex]\(y = 13.69 - 1.77 = 11.92\)[/tex]). Another point is [tex]\((1, 11.92)\)[/tex].
2. Second Line:
- Start at [tex]\((0, 5)\)[/tex].
- Use the slope [tex]\(\frac{1}{3}\)[/tex] to find another point. For example, move 3 units to the right (to [tex]\(x=3\)[/tex]), and then move up 1 unit (to [tex]\(y = 5 + 1 = 6\)[/tex]). Another point is [tex]\((3, 6)\)[/tex].
3. Draw straight lines through the points for each equation.
### Step 6: Verify the Intersection Point
Upon graphing:
- The lines will intersect at approximately [tex]\((4.13, 6.38)\)[/tex].
This confirms the coordinates of the intersection that we obtained from solving the equations. The intersection point should lie on both lines when they are graphed accordingly.
### Step 7: Plot the Intersection Point
- Plot the point [tex]\((4.13, 6.38)\)[/tex] on the graph.
This completes the process of graphing the system of equations, showing the point where they intersect.
### Step 1: Understand the System of Equations
We have two equations:
1. [tex]\( 1.15x + 0.65y = 8.90 \)[/tex]
2. [tex]\( x - 3y = -15 \)[/tex]
### Step 2: Find the Intersection Point
The solution to this system is the point where the two lines intersect. Given the solution:
[tex]\[ x = 4.13414634146341 \][/tex]
[tex]\[ y = 6.37804878048780 \][/tex]
So, the intersection point is approximately [tex]\((4.13, 6.38)\)[/tex].
### Step 3: Rewrite Each Equation in Slope-Intercept Form
To graph the lines, it's often easiest to rewrite the equations in the form [tex]\( y = mx + b \)[/tex] where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
#### For the first equation:
[tex]\[ 1.15x + 0.65y = 8.90 \][/tex]
To solve for [tex]\( y \)[/tex]:
[tex]\[ 0.65y = -1.15x + 8.90 \][/tex]
[tex]\[ y = \frac{-1.15}{0.65}x + \frac{8.90}{0.65} \][/tex]
[tex]\[ y = -1.76923x + 13.69231 \][/tex]
So, the first equation in slope-intercept form is:
[tex]\[ y = -1.76923x + 13.69231 \][/tex]
#### For the second equation:
[tex]\[ x - 3y = -15 \][/tex]
To solve for [tex]\( y \)[/tex]:
[tex]\[ -3y = -x - 15 \][/tex]
[tex]\[ y = \frac{1}{3}x + 5 \][/tex]
So, the second equation in slope-intercept form is:
[tex]\[ y = \frac{1}{3}x + 5 \][/tex]
### Step 4: Plot Each Line on the Graph
#### First Equation: [tex]\( y = -1.76923x + 13.69231 \)[/tex]
- The y-intercept ([tex]\(b\)[/tex]) is approximately 13.69, so the line crosses the y-axis at [tex]\((0, 13.69)\)[/tex].
- The slope ([tex]\(m\)[/tex]) is approximately [tex]\(-1.77\)[/tex].
- From the y-intercept, for every unit we move to the right on the x-axis, the line falls by about 1.77 units.
#### Second Equation: [tex]\( y = \frac{1}{3}x + 5 \)[/tex]
- The y-intercept ([tex]\(b\)[/tex]) is 5, so the line crosses the y-axis at [tex]\((0, 5)\)[/tex].
- The slope ([tex]\(\frac{1}{3}\)[/tex]) means that for every unit we move to the right on the x-axis, the line rises by 1/3 of a unit.
### Step 5: Graph the Lines
1. First Line:
- Start at [tex]\((0, 13.69)\)[/tex].
- Use the slope [tex]\(-1.77\)[/tex] to find another point. For example, move 1 unit to the right (to [tex]\(x=1\)[/tex]), and then move down 1.77 units (to [tex]\(y = 13.69 - 1.77 = 11.92\)[/tex]). Another point is [tex]\((1, 11.92)\)[/tex].
2. Second Line:
- Start at [tex]\((0, 5)\)[/tex].
- Use the slope [tex]\(\frac{1}{3}\)[/tex] to find another point. For example, move 3 units to the right (to [tex]\(x=3\)[/tex]), and then move up 1 unit (to [tex]\(y = 5 + 1 = 6\)[/tex]). Another point is [tex]\((3, 6)\)[/tex].
3. Draw straight lines through the points for each equation.
### Step 6: Verify the Intersection Point
Upon graphing:
- The lines will intersect at approximately [tex]\((4.13, 6.38)\)[/tex].
This confirms the coordinates of the intersection that we obtained from solving the equations. The intersection point should lie on both lines when they are graphed accordingly.
### Step 7: Plot the Intersection Point
- Plot the point [tex]\((4.13, 6.38)\)[/tex] on the graph.
This completes the process of graphing the system of equations, showing the point where they intersect.
