A System Of Equations Is Given Below.$\[ Y=\frac{1}{2} X-3 \\] $\[ Y=-\frac{1}{2} X-3 \\]Which Of The Following Statements Best Describes The Two Lines?A. They Have The Same Slope But Different \[$ Y \$\]-intercepts, So They

Introduction

When dealing with a system of equations, it's essential to understand the relationship between the two equations. In this case, we have two linear equations in the form of $y = mx + b$, where $m$ represents the slope and $b$ represents the $y$-intercept. The two equations are:

${ y = \frac{1}{2} x - 3 }$

${ y = -\frac{1}{2} x - 3 }$

To determine the relationship between these two lines, we need to analyze their slopes and $y$-intercepts.

Slope and $y$-Intercept

The slope of a line is a measure of how steep it is, and it's calculated as the ratio of the change in $y$ to the change in $x$. In the first equation, the slope is $\frac{1}{2}$, which means that for every unit increase in $x$, the value of $y$ increases by $\frac{1}{2}$ unit. In the second equation, the slope is $-\frac{1}{2}$, which means that for every unit increase in $x$, the value of $y$ decreases by $\frac{1}{2}$ unit.

The $y$-intercept is the point where the line intersects the $y$-axis. In both equations, the $y$-intercept is $-3$, which means that both lines intersect the $y$-axis at the same point.

Analyzing the Relationship Between the Two Lines

Now that we have analyzed the slopes and $y$-intercepts of the two lines, we can determine the relationship between them. Since the slopes of the two lines are different, they are not parallel. However, since the $y$-intercepts are the same, they intersect at the same point.

Conclusion

In conclusion, the two lines represented by the equations $y = \frac{1}{2} x - 3$ and $y = -\frac{1}{2} x - 3$ have the same $y$-intercept but different slopes. This means that they intersect at the same point, but they are not parallel.

Which of the Following Statements Best Describes the Two Lines?

Based on our analysis, we can conclude that the two lines have the same slope but different $y$-intercepts, so they do not intersect. However, this is not one of the options provided. The correct answer is that they have the same $y$-intercept but different slopes, so they intersect at one point.

Why Do the Two Lines Have the Same $y$-Intercept?

The two lines have the same $y$-intercept because the constant term in both equations is the same. In the first equation, the constant term is $-3$, and in the second equation, the constant term is also $-3$. This means that both lines intersect the $y$-axis at the same point.

What Does the Slope Represent?

The slope of a line represents the rate of change of the dependent variable with respect to the independent variable. In the first equation, the slope is $\frac{1}{2}$, which means that for every unit increase in $x$, the value of $y$ increases by $\frac{1}{2}$ unit. In the second equation, the slope is $-\frac{1}{2}$, which means that for every unit increase in $x$, the value of $y$ decreases by $\frac{1}{2}$ unit.

Why Do the Two Lines Have Different Slopes?

The two lines have different slopes because the coefficients of the $x$ term in both equations are different. In the first equation, the coefficient of the $x$ term is $\frac{1}{2}$, and in the second equation, the coefficient of the $x$ term is $-\frac{1}{2}$. This means that the rate of change of the dependent variable with respect to the independent variable is different for the two lines.

What Does the $y$-Intercept Represent?

The $y$-intercept of a line represents the point where the line intersects the $y$-axis. In both equations, the $y$-intercept is $-3$, which means that both lines intersect the $y$-axis at the same point.

Why Do the Two Lines Have the Same $y$-Intercept?

The two lines have the same $y$-intercept because the constant term in both equations is the same. In the first equation, the constant term is $-3$, and in the second equation, the constant term is also $-3$. This means that both lines intersect the $y$-axis at the same point.

Conclusion

In conclusion, the two lines represented by the equations $y = \frac{1}{2} x - 3$ and $y = -\frac{1}{2} x - 3$ have the same $y$-intercept but different slopes. This means that they intersect at the same point, but they are not parallel.

Final Answer

The final answer is that the two lines have the same $y$-intercept but different slopes, so they intersect at one point.

Introduction

In our previous article, we discussed the relationship between two linear equations in the form of $y = mx + b$, where $m$ represents the slope and $b$ represents the $y$-intercept. We analyzed the slopes and $y$-intercepts of the two lines and determined that they have the same $y$-intercept but different slopes. In this article, we will answer some frequently asked questions related to the topic.

Q: What is the relationship between the two lines?

A: The two lines have the same $y$-intercept but different slopes. This means that they intersect at the same point, but they are not parallel.

Q: Why do the two lines have the same $y$-intercept?

A: The two lines have the same $y$-intercept because the constant term in both equations is the same. In the first equation, the constant term is $-3$, and in the second equation, the constant term is also $-3$. This means that both lines intersect the $y$-axis at the same point.

Q: What does the slope represent?

A: The slope of a line represents the rate of change of the dependent variable with respect to the independent variable. In the first equation, the slope is $\frac{1}{2}$, which means that for every unit increase in $x$, the value of $y$ increases by $\frac{1}{2}$ unit. In the second equation, the slope is $-\frac{1}{2}$, which means that for every unit increase in $x$, the value of $y$ decreases by $\frac{1}{2}$ unit.

Q: Why do the two lines have different slopes?

A: The two lines have different slopes because the coefficients of the $x$ term in both equations are different. In the first equation, the coefficient of the $x$ term is $\frac{1}{2}$, and in the second equation, the coefficient of the $x$ term is $-\frac{1}{2}$. This means that the rate of change of the dependent variable with respect to the independent variable is different for the two lines.

Q: What does the $y$-intercept represent?

A: The $y$-intercept of a line represents the point where the line intersects the $y$-axis. In both equations, the $y$-intercept is $-3$, which means that both lines intersect the $y$-axis at the same point.

Q: Can two lines with the same $y$-intercept but different slopes intersect at more than one point?

A: No, two lines with the same $y$-intercept but different slopes can only intersect at one point.

Q: Can two lines with the same slope but different $y$-intercepts intersect at more than one point?

A: No, two lines with the same slope but different $y$-intercepts can only intersect at one point.

Q: Can two lines with the same slope and the same $y$-intercept be parallel?

A: No, two lines with the same slope and the same $y$-intercept cannot be parallel. They must intersect at the same point.

Q: Can two lines with different slopes and the same $y$-intercept be parallel?

A: No, two lines with different slopes and the same $y$-intercept cannot be parallel. They must intersect at the same point.

Conclusion

In conclusion, the relationship between two lines with the same $y$-intercept but different slopes is that they intersect at the same point, but they are not parallel. We hope that this Q&A article has helped to clarify any confusion and provide a better understanding of the topic.

Final Answer

The final answer is that two lines with the same $y$-intercept but different slopes can only intersect at one point.