Question

What can you say about two lines with the same \(x\) -intercept and the same \(y\) -intercept? Assume that the \(x\) -intercept is not \(0 .\)

Short answer

Two lines with the same x-intercept and y-intercept are the same line.

Step-by-step solution

- Understand Intercepts

The x-intercept is where the line crosses the x-axis, meaning the point where y = 0. The y-intercept is where the line crosses the y-axis, meaning the point where x = 0.

- Define the Intercepts Mathematically

Let the x-intercept be at \(a, 0\) and the y-intercept be at \(0, b\), where \( a \eq 0 \). Therefore, the known points on both lines are \( (a, 0) \) and \( (0, b) \).

- Formulate the Equations of the Lines

Use the intercept form of the line equation: \[ \frac{x}{a} + \frac{y}{b} = 1 \]. From this, rewrite the line equation in standard form: \[ bx + ay = ab \].

- Analyze the Equations

Notice that the equation \( bx + ay = ab \) is determined by the intercepts \(a\) and \(b\). Since both lines have the same x and y intercepts, they have the same equation.

- Conclusion

Since the equations of both lines are identical, the two lines are not just intersecting but are indeed the same line.

Key concepts

x-intercept

The x-intercept of a line is the point where the line crosses the x-axis. This is an important concept because it's where the y-value is zero. To find the x-intercept, we set y = 0 in the line's equation. Mathematically, if we have a line equation in the form y = mx + b, we find the x-intercept by solving the equation 0 = mx + b for x.
In simpler terms, the x-intercept is the value of x when y is zero. For instance, consider the line equation y = 2x + 6. To find the x-intercept, set y = 0, which gives us: 0 = 2x + 6. Solving for x, we get x = -3. Thus, the x-intercept is at the point (-3, 0).
Knowing how to find the x-intercept can help you understand where a line crosses the horizontal axis, which is useful in graphing the line and solving algebraic problems.

y-intercept

The y-intercept of a line is the point where the line crosses the y-axis, which means the x-coordinate is zero at this point. To find the y-intercept, set x = 0 in the equation of the line. In the equation y = mx + b, the y-intercept is simply the value of b because when x is 0, the equation simplifies to y = b.
Imagine you're working with the line equation y = 3x - 4. To find the y-intercept, set x = 0. This gives us y = -4. Therefore, the y-intercept is at the point (0, -4).
Understanding the y-intercept allows you to know where a line will cross the vertical axis. This is particularly helpful when plotting the line on a graph or when interpreting the linear relationship between variables.

line equation

A line equation is a mathematical representation of a line and describes the relationship between the x and y coordinates on a graph. It is usually written in the form y = mx + b, where m is the slope of the line and b is the y-intercept.
Another useful form of the line equation is the intercept form, which is \(\frac{x}{a} + \frac{y}{b} = 1\). Here, a is the x-intercept, and b is the y-intercept. This form can be particularly handy when you know both intercepts of the line.
Consider a line with an x-intercept (a, 0) and a y-intercept (0, b). The intercept form can be written by substituting those values: \(\frac{x}{a} + \frac{y}{b} = 1\). Rewriting it in the standard form, we get bx + ay = ab.
Understanding different forms of line equations helps you analyze and graph linear relationships efficiently. It also allows you to convert equations into a form that might simplify solving a given problem.