Question

Use the following information. The equation represents the intercept form of the equation of a line. In the equation, the \(x\) -intercept is \(a\) and the \(y\) -intercept is \(b\). $$\frac{x}{a}+\frac{y}{b}=1$$ Write the intercept form of the equation of the line whose \(x\) -intercept is 2 and \(y\) -intercept is 3

Short answer

The intercept form of the equation of the line whose \(x\)-intercept is 2 and \(y\)-intercept is 3 is \(\frac{x}{2} + \frac{y}{3} = 1\).

Step-by-step solution

Identify the Given Information

From the exercise, you know that the \(x\)-intercept is 2 and the \(y\)-intercept is 3. The equation of a line is given by \(\frac{x}{a} + \frac{y}{b} = 1\), where \(a\) is the \(x\)-intercept and \(b\) is the \(y\)-intercept. Here, \(a = 2\) and \(b = 3\).

Substitute the Given Information into the Equation

Replace \(a\) and \(b\) in the equation with the values 2 and 3 respectively. Doing this, you get \(\frac{x}{2} + \frac{y}{3} = 1\). This is the intercept form of the equation of the line whose \(x\)-intercept is 2 and \(y\)-intercept is 3.

Key concepts

X-Intercept

When trying to visualize a line on a graph, the x-intercept is a fundamental concept. It's where the line crosses the x-axis. At this point, the value of y is zero because it's on the x-axis itself. In the intercept form of a line, given by \( \frac{x}{a} + \frac{y}{b} = 1 \) where \( a \) and \( b \) are the intercepts, the x-intercept is denoted by \( a \) as seen in our example with \( a = 2 \).

To find the x-intercept from an equation, simply set \( y = 0 \) and solve for \( x \). In a practical sense, if you're told a line crosses the x-axis at 2, that means anywhere the line meets this axis, it will be at the point \( (2, 0) \), which is a clear and straightforward way to plot it on a graph.

Y-Intercept

Complementary to the x-intercept, the y-intercept is where the line crosses the y-axis. At this moment, the x-value is zero. In the equation \( \frac{x}{a} + \frac{y}{b} = 1 \), the y-intercept is represented by \( b \), which is 3 in our given problem.

Identifying the y-intercept allows you to determine one point through which the line passes, namely \( (0, b) \) on the graph. For instance, if a line has a y-intercept at 3, the exact point on the y-axis where the line would intersect is \( (0, 3) \). Understanding both the x and y-intercepts helps in sketching the full line.

Equation of a Line

The equation of a line is a mathematical way to describe the line's behavior on a coordinate plane. The intercept form, \( \frac{x}{a} + \frac{y}{b} = 1 \) is particularly useful when we already know where the line crosses the axes. In this form, \( a \) and \( b \) are directly the x and y-intercepts, which offers a quick and easy method to write the equation given this specific information.

As we've done in our example, substituting the known intercepts into the equation provides us immediately with the line's intercept form. Other forms of the line include slope-intercept form and point-slope form, each providing different advantages depending on the known parameters.

Substitution Method

The substitution method is a technique often used in algebra to solve systems of equations, but it's equally useful when working with the equation of a line. To apply this method, you replace variables with their known values.

In our step-by-step solution, substituting \( a \) and \( b \) with 2 and 3 respectively, we are able to transform the general intercept form into the specific equation of a line that fits the given conditions. This transformation is clear and simple: \( \frac{x}{2} + \frac{y}{3} = 1 \). It’s a straightforward approach to modifying an equation to match the features of the line you're working with.