Question

what type of solutions should i get based on the equations?
5)
\\(\

$$\begin{cases} y = x + 5 \\\\ y = -\\frac{3}{2}x - 5 \\end{cases}$$

\\)
type of solution:
6)
\\(\

$$\begin{cases} y = 2x - 5 \\\\ y = 2x \\end{cases}$$

\\)
type of solution:
7)
\\(\

$$\begin{cases} y = -\\frac{5}{3}x - 4 \\\\ y = -\\frac{5}{3}x - 4 \\end{cases}$$

\\)
type of solution:
convert the following equations from standard form to slope intercept form.
show all steps!
8)
\\(2x - 3y = -3\\)
9)
\\(2x - 2y = -12\\)
10)
\\(x + 4y = 4\\)

Explanation:

5) Step1: Compare slopes

Slopes: \(1\) vs \(-\frac{3}{2}\) (different)

5) Step2: Determine solution type

Different slopes → intersect once

6) Step1: Compare slopes

Slopes: \(2\) vs \(2\) (same)

6) Step2: Compare intercepts

Intercepts: \(-5\) vs \(0\) (different)

6) Step3: Determine solution type

Same slope, different intercepts → parallel lines

7) Step1: Compare equations

Both are \(y = -\frac{5}{3}x - 4\)

7) Step2: Determine solution type

Identical lines → overlap everywhere

8) Step1: Isolate \(y\) term

\(-3y = -2x - 3\)

8) Step2: Solve for \(y\)

\(y = \frac{2}{3}x + 1\)

9) Step1: Isolate \(y\) term

\(-2y = -2x -12\)

9) Step2: Solve for \(y\)

\(y = x + 6\)

10) Step1: Isolate \(y\) term

\(4y = -x + 4\)

10) Step2: Solve for \(y\)

\(y = -\frac{1}{4}x + 1\)

Answer:

1. One unique solution
2. No solution
3. Infinitely many solutions
4. \(y = \frac{2}{3}x + 1\)
5. \(y = x + 6\)
6. \(y = -\frac{1}{4}x + 1\)