to a system of linear equations
$y = x - 5$
$x = y + 5$
$\circ$ $\boldsymbol{48}$ no solution $\boldsymbol{49}$ one solution $\boldsymbol{49}$ infinitely many solutions
$y = x$
$y = 8x$
$\circ$ $\boldsymbol{48}$ no solution $\boldsymbol{49}$ one solution $\boldsymbol{49}$ infinitely many solutions
$y = x + 6$
$y = x + 1$
$\circ$ $\boldsymbol{48}$ no solution $\boldsymbol{49}$ one solution $\boldsymbol{49}$ infinitely many solutions
$y = \frac{1}{4}x + 2$
$y = -\frac{1}{4}x + 2$
$\circ$ $\boldsymbol{48}$ no solution $\boldsymbol{49}$ one solution $\boldsymbol{49}$ infinitely many solutions

Question

Accepted Answer

To determine the number of solutions for each system of linear equations, we analyze the relationships between the equations (parallel, coinciding, or intersecting lines):

### 1. System: $ y = x - 5 $ and $ x = y + 5 $
Rewrite $ x = y + 5 $ as $ y = x - 5 $ (subtract 5 from both sides). Both equations are identical (same slope, same y - intercept). Thus, the lines coincide, so there are **infinitely many solutions**.

### 2. System: $ y = x $ and $ y = 8x $
The first line has a slope of $ 1 $, and the second has a slope of $ 8 $. Different slopes mean the lines intersect at one point. Thus, there is **one solution**? Wait, no—wait, $ y = x $ and $ y = 8x $ intersect only at $ (0,0) $? Wait, no, actually, substituting $ y = x $ into $ y = 8x $ gives $ x = 8x $, so $ 7x = 0 $, so $ x = 0 $, $ y = 0 $. Wait, but the original answer in the image selected “Infinitely Many Solutions,” which is incorrect. Wait, no—wait, maybe a typo? Wait, no, $ y = x $ and $ y = 8x $: let's check slopes. Slope of $ y = x $ is $ 1 $, slope of $ y = 8x $ is $ 8 $. Different slopes, so they intersect at one point. But the image shows “Infinitely Many Solutions” selected, which is wrong. However, following the image’s selection (maybe a mistake in the problem or the image), but mathematically, it should be “One Solution.” Wait, no—wait, maybe I misread. Wait, $ y = x $ and $ y = 8x $: if we set $ x = 8x $, $ 7x = 0 $, so $ x = 0 $, $ y = 0 $. So one solution. But the image has “Infinitely Many” selected. Maybe the problem was mistyped (e.g., $ y = x $ and $ y = x $ would be infinitely many). Assuming the image’s selection is correct (maybe a typo in the problem), but mathematically, let's proceed.

### 3. System: $ y = x + 6 $ and $ y = x + 1 $
Both lines have a slope of $ 1 $ (same slope) but different y - intercepts ($ 6 $ vs. $ 1 $). Parallel lines never intersect, so there is **no solution**. But the image shows “One Solution” selected, which is wrong. Mathematically, it should be “No Solution.”

### 4. System: $ y=\frac{1}{4}x + 2 $ and $ y=-\frac{1}{4}x + 2 $
Slope of first line: $ \frac{1}{4} $, slope of second line: $ -\frac{1}{4} $ (different slopes). Different slopes mean the lines intersect at one point. Thus, there is **one solution**.

### Correcting the Image’s Mistakes (Mathematically):
1. $ y = x - 5 $ and $ x = y + 5 $: Infinitely Many Solutions (correct, as they are the same line).
2. $ y = x $ and $ y = 8x $: One Solution (image’s selection is wrong).
3. $ y = x + 6 $ and $ y = x + 1 $: No Solution (image’s selection is wrong).
4. $ y=\frac{1}{4}x + 2 $ and $ y=-\frac{1}{4}x + 2 $: One Solution (correct, as image selected).

### Final Answers (Mathematically):
1. Infinitely Many Solutions
2. One Solution
3. No Solution
4. One Solution

If we strictly follow the image’s selected options (ignoring mathematical correctness for the sake of matching the image):
1. Infinitely Many Solutions
2. Infinitely Many Solutions (incorrect mathematically)
3. One Solution (incorrect mathematically)
4. One Solution

Since the problem asks to analyze the system, the mathematical answers are:

1. For $ y = x - 5 $ and $ x = y + 5 $:
# Explanation:
## Step1: Rewrite the second equation
Rewrite $ x = y + 5 $ as $ y = x - 5 $ (subtract 5 from both sides).
## Step2: Compare with the first equation
The first equation is $ y = x - 5 $, and the rewritten second equation is also $ y = x - 5 $. So the two equations represent the same line.
# Answer: Infinitely Many Solutions

2. For $ y = x $ and $ y = 8x $:
# Explanation:
## Step1: Set the equations equal
Substitute $ y = x $ into $ y = 8x $, we get $ x = 8x $.
## Step2: Solve for $ x $
Subtract $ x $ from both sides: $ 0 = 7x $, so $ x = 0 $. Then $ y = 0 $. There is only one solution $(0,0)$.
# Answer: One Solution

3. For $ y = x + 6 $ and $ y = x + 1 $:
# Explanation:
## Step1: Analyze slopes and intercepts
Both equations are in slope - intercept form ($ y=mx + b $). The slope $ m $ for both is $ 1 $, and the y - intercepts $ b $ are $ 6 $ and $ 1 $ (different).
## Step2: Determine the relationship between lines
Lines with the same slope and different y - intercepts are parallel, so they never intersect.
# Answer: No Solution

4. For $ y=\frac{1}{4}x + 2 $ and $ y=-\frac{1}{4}x + 2 $:
# Explanation:
## Step1: Analyze slopes
The slope of $ y=\frac{1}{4}x + 2 $ is $ \frac{1}{4} $, and the slope of $ y=-\frac{1}{4}x + 2 $ is $ -\frac{1}{4} $. The slopes are different.
## Step2: Determine the number of solutions
Lines with different slopes intersect at exactly one point.
# Answer: One Solution

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QUESTION IMAGE

Question

Explanation:

1. System: \( y = x - 5 \) and \( x = y + 5 \)

2. System: \( y = x \) and \( y = 8x \)

3. System: \( y = x + 6 \) and \( y = x + 1 \)

4. System: \( y=\frac{1}{4}x + 2 \) and \( y=-\frac{1}{4}x + 2 \)

Correcting the Image’s Mistakes (Mathematically):

Final Answers (Mathematically):

Step1: Rewrite the second equation

Step2: Compare with the first equation

Step1: Set the equations equal

Step2: Solve for \( x \)

Step1: Analyze slopes and intercepts

Step2: Determine the relationship between lines

Answer: