Question

Solve the following system of linear equations by graphing. Choose the answer that matches the solution.

y = 5x + 3

y = 5x + 15

answers:

No solutions

(3 , 5)

(5 , 3)

Infinite solutions

Answers

Answer
you sure
Answered by GPT-4o mini
To solve the system of linear equations by graphing, we need to analyze the two equations:

1. \( y = 5x + 3 \)
2. \( y = 5x + 15 \)

Both equations are in slope-intercept form (y = mx + b), where the slope (m) is 5 for both lines. However, the y-intercepts (b) are different:

- The first line has a y-intercept of 3.
- The second line has a y-intercept of 15.

This means that both lines have the same slope but different y-intercepts, indicating that they are parallel lines.

When graphing, parallel lines will never intersect. Therefore, there are no points (x, y) that satisfy both equations simultaneously.

Thus, the correct answer is:

**No solutions**
Answered by GPT-4o mini
Yes, I am sure. Let's revisit the setup step by step to reinforce the conclusion:

1. **Identifying the equations**:
- Equation 1: \( y = 5x + 3 \) (this line has a slope of 5 and a y-intercept of 3)
- Equation 2: \( y = 5x + 15 \) (this line also has a slope of 5 but a different y-intercept of 15)

2. **Understanding the slopes**:
- Both lines have the same slope (5), which means they rise at the same angle.

3. **Understanding the intercepts**:
- The difference in y-intercepts (3 and 15) indicates that the two lines are parallel, meaning they will never cross or intersect at any point.

4. **Conclusion**:
- Since the lines are parallel, there is no solution that satisfies both equations simultaneously.

The final conclusion remains that there are **No solutions** to this system of equations, as they represent two distinct parallel lines in a graph.

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