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Sagot :
Sure, let's solve and plot the given linear equations step by step.
### Step-by-Step Solution:
#### 1. Express the equations:
Given the two linear equations:
[tex]\[ \begin{array}{l} y = -x - 7 \qquad \text{(Equation 1)} \\ y = x - 1 \qquad \text{(Equation 2)} \end{array} \][/tex]
#### 2. Find the Intersection Point:
To find the solution where the lines intersect, we need to solve the system of equations algebraically by setting them equal to each other because they both equal [tex]\( y \)[/tex].
[tex]\[ -x - 7 = x - 1 \][/tex]
Combine like terms:
[tex]\[ -7 + 1 = 2x \][/tex]
[tex]\[ -6 = 2x \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ x = -3 \][/tex]
Substitute [tex]\( x = -3 \)[/tex] into either of the original equations to find [tex]\( y \)[/tex]. Let's use Equation 2:
[tex]\[ y = -3 - 1 \][/tex]
[tex]\[ y = -4 \][/tex]
So, the solution to the system of equations is:
[tex]\[ (x, y) = (-3, -4) \][/tex]
#### 3. Plot the Equations and the Solution:
Now, let's plot the two lines and mark the solution point.
- For [tex]\( y = -x - 7 \)[/tex]:
- When [tex]\( x = -10 \)[/tex], [tex]\( y = -(-10) - 7 = 3 \)[/tex].
- When [tex]\( x = 0 \)[/tex], [tex]\( y = -0 - 7 = -7 \)[/tex].
- When [tex]\( x = 10 \)[/tex], [tex]\( y = -10 - 7 = -17 \)[/tex].
- For [tex]\( y = x - 1 \)[/tex]:
- When [tex]\( x = -10 \)[/tex], [tex]\( y = -10 - 1 = -11 \)[/tex].
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 0 - 1 = -1 \)[/tex].
- When [tex]\( x = 10 \)[/tex], [tex]\( y = 10 - 1 = 9 \)[/tex].
The solution point is at [tex]\( (-3, -4) \)[/tex].
#### 4. Plotting:
1. Draw the first line [tex]\( y = -x - 7 \)[/tex]:
- Plot points: [tex]\((-10, 3)\)[/tex], [tex]\((0, -7)\)[/tex], [tex]\((10, -17)\)[/tex].
- Connect these points with a straight line.
2. Draw the second line [tex]\( y = x - 1 \)[/tex]:
- Plot points: [tex]\((-10, -11)\)[/tex], [tex]\((0, -1)\)[/tex], [tex]\( (10, 9)\)[/tex].
- Connect these points with a straight line.
3. Mark the intersection (solution) point [tex]\( (-3, -4) \)[/tex] on the graph:
- Plot this point and label it as the solution.
### Final Plot:
```
\ |
|
-y = -x -7
--+
-x -3| Solution (-3, -4)
|
--1|
- -x
\ | x - 1
```
I hope this step-by-step solution helps you understand how to solve and plot the given system of linear equations!
### Step-by-Step Solution:
#### 1. Express the equations:
Given the two linear equations:
[tex]\[ \begin{array}{l} y = -x - 7 \qquad \text{(Equation 1)} \\ y = x - 1 \qquad \text{(Equation 2)} \end{array} \][/tex]
#### 2. Find the Intersection Point:
To find the solution where the lines intersect, we need to solve the system of equations algebraically by setting them equal to each other because they both equal [tex]\( y \)[/tex].
[tex]\[ -x - 7 = x - 1 \][/tex]
Combine like terms:
[tex]\[ -7 + 1 = 2x \][/tex]
[tex]\[ -6 = 2x \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ x = -3 \][/tex]
Substitute [tex]\( x = -3 \)[/tex] into either of the original equations to find [tex]\( y \)[/tex]. Let's use Equation 2:
[tex]\[ y = -3 - 1 \][/tex]
[tex]\[ y = -4 \][/tex]
So, the solution to the system of equations is:
[tex]\[ (x, y) = (-3, -4) \][/tex]
#### 3. Plot the Equations and the Solution:
Now, let's plot the two lines and mark the solution point.
- For [tex]\( y = -x - 7 \)[/tex]:
- When [tex]\( x = -10 \)[/tex], [tex]\( y = -(-10) - 7 = 3 \)[/tex].
- When [tex]\( x = 0 \)[/tex], [tex]\( y = -0 - 7 = -7 \)[/tex].
- When [tex]\( x = 10 \)[/tex], [tex]\( y = -10 - 7 = -17 \)[/tex].
- For [tex]\( y = x - 1 \)[/tex]:
- When [tex]\( x = -10 \)[/tex], [tex]\( y = -10 - 1 = -11 \)[/tex].
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 0 - 1 = -1 \)[/tex].
- When [tex]\( x = 10 \)[/tex], [tex]\( y = 10 - 1 = 9 \)[/tex].
The solution point is at [tex]\( (-3, -4) \)[/tex].
#### 4. Plotting:
1. Draw the first line [tex]\( y = -x - 7 \)[/tex]:
- Plot points: [tex]\((-10, 3)\)[/tex], [tex]\((0, -7)\)[/tex], [tex]\((10, -17)\)[/tex].
- Connect these points with a straight line.
2. Draw the second line [tex]\( y = x - 1 \)[/tex]:
- Plot points: [tex]\((-10, -11)\)[/tex], [tex]\((0, -1)\)[/tex], [tex]\( (10, 9)\)[/tex].
- Connect these points with a straight line.
3. Mark the intersection (solution) point [tex]\( (-3, -4) \)[/tex] on the graph:
- Plot this point and label it as the solution.
### Final Plot:
```
\ |
|
-y = -x -7
--+
-x -3| Solution (-3, -4)
|
--1|
- -x
\ | x - 1
```
I hope this step-by-step solution helps you understand how to solve and plot the given system of linear equations!
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