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Sagot :
To solve the system of equations:
[tex]\[ \begin{cases} x + y = -1 \\ x - 2y = 14 \end{cases} \][/tex]
we will use the method of graphing. Here are the steps, laid out clearly:
### Step 1: Transform Equations to Slope-Intercept Form
Convert each equation to the form [tex]\( y = mx + b \)[/tex] (where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept):
1. First equation: [tex]\( x + y = -1 \)[/tex]
Subtract [tex]\( x \)[/tex] from both sides:
[tex]\[ y = -x - 1 \][/tex]
2. Second equation: [tex]\( x - 2y = 14 \)[/tex]
Subtract [tex]\( x \)[/tex] from both sides:
[tex]\[ -2y = -x + 14 \][/tex]
Divide by -2:
[tex]\[ y = \frac{x}{2} - 7 \][/tex]
So, we have:
[tex]\[ y = -x - 1 \][/tex] and [tex]\[ y = \frac{x}{2} - 7 \][/tex]
### Step 2: Plot the Lines
Next, we will plot these two lines on the same graph:
1. First line: [tex]\( y = -x - 1 \)[/tex]
- Slope [tex]\( m = -1 \)[/tex]
- Y-intercept = -1 (the point where the line crosses the y-axis)
2. Second line: [tex]\( y = \frac{x}{2} - 7 \)[/tex]
- Slope [tex]\( m = \frac{1}{2} \)[/tex]
- Y-intercept = -7 (the point where the line crosses the y-axis)
### Step 3: Find Intersection Point
The solution to the system is the point where the two lines intersect. Instead of graphing these manually, we can also solve algebraically to pinpoint the intersection exactly.
### Step 4: Algebraic Solution
We solve the system algebraically:
1. [tex]\( x + y = -1 \)[/tex] (Equation 1)
2. [tex]\( x - 2y = 14 \)[/tex] (Equation 2)
Substitute [tex]\( y \)[/tex] from Equation 1 ([tex]\( y = -x - 1 \)[/tex]) into Equation 2:
[tex]\[ x - 2(-x - 1) = 14 \][/tex]
Simplify:
[tex]\[ x + 2x + 2 = 14 \][/tex]
Combine like terms:
[tex]\[ 3x + 2 = 14 \][/tex]
Subtract 2 from both sides:
[tex]\[ 3x = 12 \][/tex]
Divide by 3:
[tex]\[ x = 4 \][/tex]
Then, substitute [tex]\( x = 4 \)[/tex] into the expression for [tex]\( y \)[/tex]:
[tex]\[ y = -x - 1 \][/tex]
[tex]\[ y = -4 - 1 = -5 \][/tex]
Thus, the intersection point, and the solution to the system of equations, is:
[tex]\[ (4, -5) \][/tex]
### Verification:
To ensure our solution is correct, we can substitute [tex]\( x = 4 \)[/tex] and [tex]\( y = -5 \)[/tex] back into both original equations:
1. [tex]\( 4 + (-5) = -1 \)[/tex]:
[tex]\[ -1 = -1 \][/tex] (True)
2. [tex]\( 4 - 2(-5) = 14 \)[/tex]:
[tex]\[ 4 + 10 = 14 \][/tex]
[tex]\[ 14 = 14 \][/tex] (True)
Both equations are satisfied, confirming that the solution [tex]\((4, -5)\)[/tex] is correct.
[tex]\[ \begin{cases} x + y = -1 \\ x - 2y = 14 \end{cases} \][/tex]
we will use the method of graphing. Here are the steps, laid out clearly:
### Step 1: Transform Equations to Slope-Intercept Form
Convert each equation to the form [tex]\( y = mx + b \)[/tex] (where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept):
1. First equation: [tex]\( x + y = -1 \)[/tex]
Subtract [tex]\( x \)[/tex] from both sides:
[tex]\[ y = -x - 1 \][/tex]
2. Second equation: [tex]\( x - 2y = 14 \)[/tex]
Subtract [tex]\( x \)[/tex] from both sides:
[tex]\[ -2y = -x + 14 \][/tex]
Divide by -2:
[tex]\[ y = \frac{x}{2} - 7 \][/tex]
So, we have:
[tex]\[ y = -x - 1 \][/tex] and [tex]\[ y = \frac{x}{2} - 7 \][/tex]
### Step 2: Plot the Lines
Next, we will plot these two lines on the same graph:
1. First line: [tex]\( y = -x - 1 \)[/tex]
- Slope [tex]\( m = -1 \)[/tex]
- Y-intercept = -1 (the point where the line crosses the y-axis)
2. Second line: [tex]\( y = \frac{x}{2} - 7 \)[/tex]
- Slope [tex]\( m = \frac{1}{2} \)[/tex]
- Y-intercept = -7 (the point where the line crosses the y-axis)
### Step 3: Find Intersection Point
The solution to the system is the point where the two lines intersect. Instead of graphing these manually, we can also solve algebraically to pinpoint the intersection exactly.
### Step 4: Algebraic Solution
We solve the system algebraically:
1. [tex]\( x + y = -1 \)[/tex] (Equation 1)
2. [tex]\( x - 2y = 14 \)[/tex] (Equation 2)
Substitute [tex]\( y \)[/tex] from Equation 1 ([tex]\( y = -x - 1 \)[/tex]) into Equation 2:
[tex]\[ x - 2(-x - 1) = 14 \][/tex]
Simplify:
[tex]\[ x + 2x + 2 = 14 \][/tex]
Combine like terms:
[tex]\[ 3x + 2 = 14 \][/tex]
Subtract 2 from both sides:
[tex]\[ 3x = 12 \][/tex]
Divide by 3:
[tex]\[ x = 4 \][/tex]
Then, substitute [tex]\( x = 4 \)[/tex] into the expression for [tex]\( y \)[/tex]:
[tex]\[ y = -x - 1 \][/tex]
[tex]\[ y = -4 - 1 = -5 \][/tex]
Thus, the intersection point, and the solution to the system of equations, is:
[tex]\[ (4, -5) \][/tex]
### Verification:
To ensure our solution is correct, we can substitute [tex]\( x = 4 \)[/tex] and [tex]\( y = -5 \)[/tex] back into both original equations:
1. [tex]\( 4 + (-5) = -1 \)[/tex]:
[tex]\[ -1 = -1 \][/tex] (True)
2. [tex]\( 4 - 2(-5) = 14 \)[/tex]:
[tex]\[ 4 + 10 = 14 \][/tex]
[tex]\[ 14 = 14 \][/tex] (True)
Both equations are satisfied, confirming that the solution [tex]\((4, -5)\)[/tex] is correct.
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