Question: 2) Solve each linear system using the . . .
4) What are the different possibilities for the number of solutions to a linear system. Explain each scenario and provide a diagram. [3]
5) Analyze the solution to the linear system given. The method of substitution was used. Is it correct? If so, how does the check prove that. Is it wrong? What was done wrong and is there a problem with the check? [2]
$$\ell_1: y = 3x - 2$$
$$\ell_2: 4x - y = -5$$
$$\ell_1: y = 3x - 2$$
Sub $$x = -3$$ into $$\ell_1$$
$$y = 3(-3) - 2$$
$$y = -9 - 2$$
$$y = -11$$
$$\ell_2: 4x - y = -5$$
Sub $$y = 3x - 2$$ into $$\ell_2$$
$$4x - 3x - 2 = -5$$
$$x - 2 = -5$$
$$x = -5 + 2$$
$$x = -3$$
Check Solution $$x = -3, y = -11$$
LS RS
= y = 3x - 2
= -11 = 3(-3) - 2
= -11
LS = RS therefore, it is the correct solution
1. **State the problem:**
We are given the linear system:
$$\ell_1: y = 3x - 2$$
$$\ell_2: 4x - y = -5$$
We need to analyze the solution found by substitution and verify its correctness.
2. **Recall substitution method:**
Substitution involves solving one equation for one variable and substituting into the other equation.
3. **Substitute $$x = -3$$ into $$\ell_1$$:**
$$y = 3(-3) - 2$$
$$y = -9 - 2$$
$$y = -11$$
4. **Substitute $$y = 3x - 2$$ into $$\ell_2$$:**
$$4x - y = -5$$
Replace $$y$$:
$$4x - (3x - 2) = -5$$
Simplify:
$$4x - 3x + 2 = -5$$
$$x + 2 = -5$$
Subtract 2 from both sides:
$$x + \cancel{2} - \cancel{2} = -5 - 2$$
$$x = -7$$
5. **Check for consistency:**
The substitution step gave $$x = -3$$ from the second equation, but simplifying carefully shows $$x = -7$$.
6. **Identify the error:**
In the original substitution into $$\ell_2$$, the step was:
$$4x - 3x - 2 = -5$$
which is incorrect because substituting $$y = 3x - 2$$ into $$4x - y = -5$$ should be:
$$4x - (3x - 2) = -5$$
which simplifies to:
$$4x - 3x + 2 = -5$$
not $$4x - 3x - 2 = -5$$.
7. **Correct solution:**
From the correct substitution:
$$x + 2 = -5$$
$$x = -7$$
Then from $$\ell_1$$:
$$y = 3(-7) - 2 = -21 - 2 = -23$$
8. **Check the solution $$x = -7, y = -23$$:**
Check in $$\ell_2$$:
$$4(-7) - (-23) = -28 + 23 = -5$$
which matches the right side.
9. **Conclusion:**
The original substitution was incorrect due to a sign error in substitution.
The check was done on the incorrect solution, so it falsely confirmed correctness.
The correct solution is $$x = -7, y = -23$$.
---
10. **Different possibilities for the number of solutions to a linear system:**
- **One solution:** The lines intersect at exactly one point (consistent and independent).
- **No solution:** The lines are parallel and distinct (inconsistent).
- **Infinite solutions:** The lines coincide (dependent).
11. **Diagram:**