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Sagot :
To solve the given system of equations and identify the solution graphically, let’s proceed step-by-step.
The system of equations is:
[tex]\[ \begin{cases} -x + y = -2 \\ 2x + y = 10 \\ \end{cases} \][/tex]
### Step 1: Rewrite each equation in slope-intercept form (y = mx + b)
1. For the first equation [tex]\(-x + y = -2\)[/tex]:
[tex]\[ y = x - 2 \quad \text{(Equation 1)} \][/tex]
2. For the second equation [tex]\(2x + y = 10\)[/tex]:
[tex]\[ y = -2x + 10 \quad \text{(Equation 2)} \][/tex]
### Step 2: Graph both equations on the same coordinate plane
#### Graphing Equation 1: [tex]\(y = x - 2\)[/tex]
- Intercept (0, -2): The y-intercept is -2.
- Slope 1: Move up 1 unit and to the right 1 unit from the y-intercept.
Plot these points and draw a line through them.
#### Graphing Equation 2: [tex]\(y = -2x + 10\)[/tex]
- Intercept (0, 10): The y-intercept is 10.
- Slope -2: Move down 2 units and to the right 1 unit from the y-intercept.
Plot these points and draw a line through them.
### Step 3: Identify the point of intersection
The solution to the system of equations is the point where the two lines intersect.
### Step 4: Solve the equations algebraically
Since graphing might not always be precise, we compute the exact intersection point algebraically:
1. Set the equations equal to each other since both equal [tex]\(y\)[/tex]:
[tex]\[ x - 2 = -2x + 10 \][/tex]
2. Solve for [tex]\(x\)[/tex]:
[tex]\[ x + 2x = 10 + 2 \][/tex]
[tex]\[ 3x = 12 \][/tex]
[tex]\[ x = 4 \][/tex]
3. Substitute [tex]\(x = 4\)[/tex] back into one of the original equations to solve for [tex]\(y\)[/tex]:
[tex]\[ y = x - 2 \rightarrow y = 4 - 2 \rightarrow y = 2 \][/tex]
So, the intersection point is [tex]\((4, 2)\)[/tex].
### Step 5: Verify by substituting [tex]\((4, 2)\)[/tex] into both equations
1. First equation: [tex]\(-x + y = -2 \rightarrow -4 + 2 = -2\)[/tex]
2. Second equation: [tex]\(2x + y = 10 \rightarrow 2(4) + 2 = 8 + 2 = 10\)[/tex]
Since [tex]\((4, 2)\)[/tex] satisfies both equations, this is the correct solution.
### Answer:
The solution to the system of equations [tex]\(-x + y = -2\)[/tex] and [tex]\(2x + y = 10\)[/tex] is [tex]\((4, 2)\)[/tex]. Graphing these lines would show their intersection at the point [tex]\((4, 2)\)[/tex]. This means [tex]\((4, 2)\)[/tex] is the correct answer.
The system of equations is:
[tex]\[ \begin{cases} -x + y = -2 \\ 2x + y = 10 \\ \end{cases} \][/tex]
### Step 1: Rewrite each equation in slope-intercept form (y = mx + b)
1. For the first equation [tex]\(-x + y = -2\)[/tex]:
[tex]\[ y = x - 2 \quad \text{(Equation 1)} \][/tex]
2. For the second equation [tex]\(2x + y = 10\)[/tex]:
[tex]\[ y = -2x + 10 \quad \text{(Equation 2)} \][/tex]
### Step 2: Graph both equations on the same coordinate plane
#### Graphing Equation 1: [tex]\(y = x - 2\)[/tex]
- Intercept (0, -2): The y-intercept is -2.
- Slope 1: Move up 1 unit and to the right 1 unit from the y-intercept.
Plot these points and draw a line through them.
#### Graphing Equation 2: [tex]\(y = -2x + 10\)[/tex]
- Intercept (0, 10): The y-intercept is 10.
- Slope -2: Move down 2 units and to the right 1 unit from the y-intercept.
Plot these points and draw a line through them.
### Step 3: Identify the point of intersection
The solution to the system of equations is the point where the two lines intersect.
### Step 4: Solve the equations algebraically
Since graphing might not always be precise, we compute the exact intersection point algebraically:
1. Set the equations equal to each other since both equal [tex]\(y\)[/tex]:
[tex]\[ x - 2 = -2x + 10 \][/tex]
2. Solve for [tex]\(x\)[/tex]:
[tex]\[ x + 2x = 10 + 2 \][/tex]
[tex]\[ 3x = 12 \][/tex]
[tex]\[ x = 4 \][/tex]
3. Substitute [tex]\(x = 4\)[/tex] back into one of the original equations to solve for [tex]\(y\)[/tex]:
[tex]\[ y = x - 2 \rightarrow y = 4 - 2 \rightarrow y = 2 \][/tex]
So, the intersection point is [tex]\((4, 2)\)[/tex].
### Step 5: Verify by substituting [tex]\((4, 2)\)[/tex] into both equations
1. First equation: [tex]\(-x + y = -2 \rightarrow -4 + 2 = -2\)[/tex]
2. Second equation: [tex]\(2x + y = 10 \rightarrow 2(4) + 2 = 8 + 2 = 10\)[/tex]
Since [tex]\((4, 2)\)[/tex] satisfies both equations, this is the correct solution.
### Answer:
The solution to the system of equations [tex]\(-x + y = -2\)[/tex] and [tex]\(2x + y = 10\)[/tex] is [tex]\((4, 2)\)[/tex]. Graphing these lines would show their intersection at the point [tex]\((4, 2)\)[/tex]. This means [tex]\((4, 2)\)[/tex] is the correct answer.
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