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Sagot :
Certainly! Let's solve and graph the given system of equations:
The given system of equations is:
[tex]\[y = \frac{4}{3} x + 4\][/tex]
[tex]\[3y = -2x - 6\][/tex]
### Step-by-Step Solution:
1. Simplify the second equation:
[tex]\[ 3y = -2x - 6 \][/tex]
Divide both sides by 3 to solve for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{-2}{3}x - 2 \][/tex]
2. Graph the first equation [tex]\( y = \frac{4}{3}x + 4 \)[/tex]:
- This is a line equation in slope-intercept form [tex]\( y = mx + b \)[/tex], where the slope [tex]\( m = \frac{4}{3} \)[/tex] and the y-intercept [tex]\( b = 4 \)[/tex].
- Start by plotting the y-intercept at [tex]\( (0, 4) \)[/tex].
- Use the slope to find another point: move up 4 units and right 3 units from the y-intercept to get the next point at [tex]\( (3, \frac{30}{3}+4) = (3, 8) \)[/tex].
- Draw the line through these points.
3. Graph the second equation [tex]\( y = \frac{-2}{3}x - 2 \)[/tex]:
- Similarly, this is in slope-intercept form where the slope [tex]\( m = \frac{-2}{3} \)[/tex] and the y-intercept [tex]\( b = -2 \)[/tex].
- Start by plotting the y-intercept at [tex]\( (0, -2) \)[/tex].
- Use the slope to find another point: move down 2 units and right 3 units from the y-intercept to get the next point at [tex]\( (3, \frac{-12}{3}-2) = (3, -4) \)[/tex].
- Draw the line through these points.
4. Find the intersection point:
- Set the two equations equal to each other to find the point of intersection:
[tex]\[ \frac{4}{3} x + 4 = \frac{-2}{3} x - 2 \][/tex]
- Combine like terms by adding [tex]\(\frac{2}{3}x\)[/tex] to both sides:
[tex]\[ \frac{6}{3}x + 4 = -2 \][/tex]
- Simplify:
[tex]\[ 2x + 4 = -2 \][/tex]
- Solve for [tex]\( x \)[/tex]:
[tex]\[ 2x = -6 \quad \Rightarrow \quad x = -3 \][/tex]
- Substitute [tex]\( x = -3 \)[/tex] back into one of the equations to find [tex]\( y \)[/tex]:
[tex]\[ y = \frac{4}{3}(-3) + 4 = -4 + 4 = 0 \][/tex]
- Therefore, the point of intersection is [tex]\( (-3, 0) \)[/tex].
### Final Answer:
- Plot the point of intersection [tex]\( (-3, 0) \)[/tex] on the graph.
- This point is where the two lines intersect and is the solution to the system.
#### To fully visualize:
- Draw each line on a coordinate plane.
- Use the mentioned drawing tools:
- Mark Feature Tool: Use this to label the intersection point [tex]\( (-3, 0) \)[/tex].
- Line Tool: Use this to draw the lines as discussed.
Here is a graphical representation:
[tex]\[ \begin{array}{l} \text{Graph the lines based on the steps above.} \\ \text{Identify the intersection point at } (-3,0). \end{array} \][/tex]
### Finished graph:
- The graph should show two intersecting lines with the intersection marked at [tex]\((-3, 0)\)[/tex].
Now, you can mark [tex]\( (-3, 0) \)[/tex] with the "Mark Feature" tool and draw the respective lines.
The given system of equations is:
[tex]\[y = \frac{4}{3} x + 4\][/tex]
[tex]\[3y = -2x - 6\][/tex]
### Step-by-Step Solution:
1. Simplify the second equation:
[tex]\[ 3y = -2x - 6 \][/tex]
Divide both sides by 3 to solve for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{-2}{3}x - 2 \][/tex]
2. Graph the first equation [tex]\( y = \frac{4}{3}x + 4 \)[/tex]:
- This is a line equation in slope-intercept form [tex]\( y = mx + b \)[/tex], where the slope [tex]\( m = \frac{4}{3} \)[/tex] and the y-intercept [tex]\( b = 4 \)[/tex].
- Start by plotting the y-intercept at [tex]\( (0, 4) \)[/tex].
- Use the slope to find another point: move up 4 units and right 3 units from the y-intercept to get the next point at [tex]\( (3, \frac{30}{3}+4) = (3, 8) \)[/tex].
- Draw the line through these points.
3. Graph the second equation [tex]\( y = \frac{-2}{3}x - 2 \)[/tex]:
- Similarly, this is in slope-intercept form where the slope [tex]\( m = \frac{-2}{3} \)[/tex] and the y-intercept [tex]\( b = -2 \)[/tex].
- Start by plotting the y-intercept at [tex]\( (0, -2) \)[/tex].
- Use the slope to find another point: move down 2 units and right 3 units from the y-intercept to get the next point at [tex]\( (3, \frac{-12}{3}-2) = (3, -4) \)[/tex].
- Draw the line through these points.
4. Find the intersection point:
- Set the two equations equal to each other to find the point of intersection:
[tex]\[ \frac{4}{3} x + 4 = \frac{-2}{3} x - 2 \][/tex]
- Combine like terms by adding [tex]\(\frac{2}{3}x\)[/tex] to both sides:
[tex]\[ \frac{6}{3}x + 4 = -2 \][/tex]
- Simplify:
[tex]\[ 2x + 4 = -2 \][/tex]
- Solve for [tex]\( x \)[/tex]:
[tex]\[ 2x = -6 \quad \Rightarrow \quad x = -3 \][/tex]
- Substitute [tex]\( x = -3 \)[/tex] back into one of the equations to find [tex]\( y \)[/tex]:
[tex]\[ y = \frac{4}{3}(-3) + 4 = -4 + 4 = 0 \][/tex]
- Therefore, the point of intersection is [tex]\( (-3, 0) \)[/tex].
### Final Answer:
- Plot the point of intersection [tex]\( (-3, 0) \)[/tex] on the graph.
- This point is where the two lines intersect and is the solution to the system.
#### To fully visualize:
- Draw each line on a coordinate plane.
- Use the mentioned drawing tools:
- Mark Feature Tool: Use this to label the intersection point [tex]\( (-3, 0) \)[/tex].
- Line Tool: Use this to draw the lines as discussed.
Here is a graphical representation:
[tex]\[ \begin{array}{l} \text{Graph the lines based on the steps above.} \\ \text{Identify the intersection point at } (-3,0). \end{array} \][/tex]
### Finished graph:
- The graph should show two intersecting lines with the intersection marked at [tex]\((-3, 0)\)[/tex].
Now, you can mark [tex]\( (-3, 0) \)[/tex] with the "Mark Feature" tool and draw the respective lines.
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