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Sagot :
Let's follow a step-by-step process to graph the system of equations and identify the solution.
### Step 1: Understand the Equations
We have two linear equations:
1. [tex]\( y = 2x - 4 \)[/tex]
2. [tex]\( y = -\frac{1}{2}x + 1 \)[/tex]
### Step 2: Graph the First Equation
To graph [tex]\( y = 2x - 4 \)[/tex]:
#### Find the y-intercept and slope:
- The y-intercept is -4 (the point where the line crosses the y-axis: (0, -4)).
- The slope is 2, which means the line rises 2 units for every 1 unit it moves to the right.
#### Plot points and draw the line:
- Start at (0, -4).
- From (0, -4), move up 2 units and to the right 1 unit to get the next point (1, -2).
- Repeat this step to plot a few more points.
- Draw a straight line through these points.
### Step 3: Graph the Second Equation
To graph [tex]\( y = -\frac{1}{2}x + 1 \)[/tex]:
#### Find the y-intercept and slope:
- The y-intercept is 1 (the point where the line crosses the y-axis: (0, 1)).
- The slope is -[tex]\(\frac{1}{2}\)[/tex], which means the line falls 1 unit for every 2 units it moves to the right.
#### Plot points and draw the line:
- Start at (0, 1).
- From (0, 1), move down 1 unit and to the right 2 units to get the next point (2, 0).
- Repeat this step to plot a few more points.
- Draw a straight line through these points.
### Step 4: Identify the Solution
Now that both lines are graphed, the point where they intersect (cross) is the solution to the system of equations.
#### Verify the Solution
- Point: (2, 0)
- Substitute x = 2 into both equations.
For [tex]\( y = 2x - 4 \)[/tex]:
[tex]\[ y = 2(2) - 4 = 4 - 4 = 0 \][/tex]
The point (2, 0) satisfies this equation.
For [tex]\( y = -\frac{1}{2}x + 1 \)[/tex]:
[tex]\[ y = -\frac{1}{2}(2) + 1 = -1 + 1 = 0 \][/tex]
The point (2, 0) also satisfies this equation.
### Step 5: Check the Given Points
We know:
- [tex]\((3, 2)\)[/tex] does not satisfy both equations.
- [tex]\((2, 3)\)[/tex] does not satisfy both equations.
- [tex]\((0, 2)\)[/tex] does not satisfy both equations.
- [tex]\((2, 0)\)[/tex] is the solution point.
### Conclusion
The solution to the system of equations [tex]\( y = 2x - 4 \)[/tex] and [tex]\( y = -\frac{1}{2}x + 1 \)[/tex] is the point where the lines intersect: [tex]\((2, 0)\)[/tex]. This point lies on both lines. So, the answer is [tex]\((2, 0)\)[/tex].
### Step 1: Understand the Equations
We have two linear equations:
1. [tex]\( y = 2x - 4 \)[/tex]
2. [tex]\( y = -\frac{1}{2}x + 1 \)[/tex]
### Step 2: Graph the First Equation
To graph [tex]\( y = 2x - 4 \)[/tex]:
#### Find the y-intercept and slope:
- The y-intercept is -4 (the point where the line crosses the y-axis: (0, -4)).
- The slope is 2, which means the line rises 2 units for every 1 unit it moves to the right.
#### Plot points and draw the line:
- Start at (0, -4).
- From (0, -4), move up 2 units and to the right 1 unit to get the next point (1, -2).
- Repeat this step to plot a few more points.
- Draw a straight line through these points.
### Step 3: Graph the Second Equation
To graph [tex]\( y = -\frac{1}{2}x + 1 \)[/tex]:
#### Find the y-intercept and slope:
- The y-intercept is 1 (the point where the line crosses the y-axis: (0, 1)).
- The slope is -[tex]\(\frac{1}{2}\)[/tex], which means the line falls 1 unit for every 2 units it moves to the right.
#### Plot points and draw the line:
- Start at (0, 1).
- From (0, 1), move down 1 unit and to the right 2 units to get the next point (2, 0).
- Repeat this step to plot a few more points.
- Draw a straight line through these points.
### Step 4: Identify the Solution
Now that both lines are graphed, the point where they intersect (cross) is the solution to the system of equations.
#### Verify the Solution
- Point: (2, 0)
- Substitute x = 2 into both equations.
For [tex]\( y = 2x - 4 \)[/tex]:
[tex]\[ y = 2(2) - 4 = 4 - 4 = 0 \][/tex]
The point (2, 0) satisfies this equation.
For [tex]\( y = -\frac{1}{2}x + 1 \)[/tex]:
[tex]\[ y = -\frac{1}{2}(2) + 1 = -1 + 1 = 0 \][/tex]
The point (2, 0) also satisfies this equation.
### Step 5: Check the Given Points
We know:
- [tex]\((3, 2)\)[/tex] does not satisfy both equations.
- [tex]\((2, 3)\)[/tex] does not satisfy both equations.
- [tex]\((0, 2)\)[/tex] does not satisfy both equations.
- [tex]\((2, 0)\)[/tex] is the solution point.
### Conclusion
The solution to the system of equations [tex]\( y = 2x - 4 \)[/tex] and [tex]\( y = -\frac{1}{2}x + 1 \)[/tex] is the point where the lines intersect: [tex]\((2, 0)\)[/tex]. This point lies on both lines. So, the answer is [tex]\((2, 0)\)[/tex].
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