IDNLearn.com: Your trusted source for finding accurate answers. Discover detailed answers to your questions with our extensive database of expert knowledge.
Sagot :
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.
We are happy to have you as part of our community. Keep asking, answering, and sharing your insights. Together, we can create a valuable knowledge resource. Thank you for trusting IDNLearn.com. We’re dedicated to providing accurate answers, so visit us again for more solutions.
