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Sagot :
To solve the system of equations:
[tex]\[ \begin{array}{c} y = -2x + 3 \\ y = x - 6 \end{array} \][/tex]
we will graph both equations and find their points of intersection.
### Step 1: Graph [tex]\( y = -2x + 3 \)[/tex]
This is a straight line with a slope of -2 and a y-intercept of 3.
- Y-Intercept: When [tex]\( x = 0 \)[/tex], [tex]\( y = -2(0) + 3 = 3 \)[/tex]. So the point (0, 3) is on the graph.
- Slope: For every increase of 1 in [tex]\( x \)[/tex], [tex]\( y \)[/tex] decreases by 2. So another point can be found by starting at (0, 3) and moving right 1 unit ([tex]\( x = 1 \)[/tex]) and down 2 units ([tex]\( y = 1 \)[/tex]). This gives the point (1, 1).
### Step 2: Graph [tex]\( y = x - 6 \)[/tex]
This is a straight line with a slope of 1 and a y-intercept of -6.
- Y-Intercept: When [tex]\( x = 0 \)[/tex], [tex]\( y = 0 - 6 = -6 \)[/tex]. So the point (0, -6) is on the graph.
- Slope: For every increase of 1 in [tex]\( x \)[/tex], [tex]\( y \)[/tex] also increases by 1. So another point can be found by starting at (0, -6) and moving right 1 unit ([tex]\( x = 1 \)[/tex]) and up 1 unit ([tex]\( y = -5 \)[/tex]). This gives the point (1, -5).
### Step 3: Find the Intersection
We solve the equations algebraically to find where they intersect graphically:
[tex]\[ -2x + 3 = x - 6 \][/tex]
Combine like terms:
[tex]\[ 3 + 6 = x + 2x \][/tex]
Simplify:
[tex]\[ 9 = 3x \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ x = 3 \][/tex]
Substitute [tex]\( x = 3 \)[/tex] back into one of the original equations to solve for [tex]\( y \)[/tex]:
[tex]\[ y = -2(3) + 3 = -6 + 3 = -3 \][/tex]
Thus, the intersection point is [tex]\( (3, -3) \)[/tex].
### Conclusion
Both lines [tex]\( y = -2x + 3 \)[/tex] and [tex]\( y = x - 6 \)[/tex] intersect at exactly one point, which is [tex]\( (3, -3) \)[/tex].
The correct answer is:
[tex]\[ \text{one solution: } \{3,-3\} \][/tex]
[tex]\[ \begin{array}{c} y = -2x + 3 \\ y = x - 6 \end{array} \][/tex]
we will graph both equations and find their points of intersection.
### Step 1: Graph [tex]\( y = -2x + 3 \)[/tex]
This is a straight line with a slope of -2 and a y-intercept of 3.
- Y-Intercept: When [tex]\( x = 0 \)[/tex], [tex]\( y = -2(0) + 3 = 3 \)[/tex]. So the point (0, 3) is on the graph.
- Slope: For every increase of 1 in [tex]\( x \)[/tex], [tex]\( y \)[/tex] decreases by 2. So another point can be found by starting at (0, 3) and moving right 1 unit ([tex]\( x = 1 \)[/tex]) and down 2 units ([tex]\( y = 1 \)[/tex]). This gives the point (1, 1).
### Step 2: Graph [tex]\( y = x - 6 \)[/tex]
This is a straight line with a slope of 1 and a y-intercept of -6.
- Y-Intercept: When [tex]\( x = 0 \)[/tex], [tex]\( y = 0 - 6 = -6 \)[/tex]. So the point (0, -6) is on the graph.
- Slope: For every increase of 1 in [tex]\( x \)[/tex], [tex]\( y \)[/tex] also increases by 1. So another point can be found by starting at (0, -6) and moving right 1 unit ([tex]\( x = 1 \)[/tex]) and up 1 unit ([tex]\( y = -5 \)[/tex]). This gives the point (1, -5).
### Step 3: Find the Intersection
We solve the equations algebraically to find where they intersect graphically:
[tex]\[ -2x + 3 = x - 6 \][/tex]
Combine like terms:
[tex]\[ 3 + 6 = x + 2x \][/tex]
Simplify:
[tex]\[ 9 = 3x \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ x = 3 \][/tex]
Substitute [tex]\( x = 3 \)[/tex] back into one of the original equations to solve for [tex]\( y \)[/tex]:
[tex]\[ y = -2(3) + 3 = -6 + 3 = -3 \][/tex]
Thus, the intersection point is [tex]\( (3, -3) \)[/tex].
### Conclusion
Both lines [tex]\( y = -2x + 3 \)[/tex] and [tex]\( y = x - 6 \)[/tex] intersect at exactly one point, which is [tex]\( (3, -3) \)[/tex].
The correct answer is:
[tex]\[ \text{one solution: } \{3,-3\} \][/tex]
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