Problem 16 - Algebra 2

Question

Write an equation for each line. \(x\) -intercept \(-2, y\) -intercept \(-6\)

Step-by-Step Solution

Answer

The equation of the line with x-intercept at -2 and y-intercept at -6 is 3x + y + 6 = 0.

Step 1: Understand the intercepts

An x-intercept is a point where the line crosses the x-axis and the y-coordinate is 0. Similarly, a y-intercept is a point where the line crosses the y-axis and the x-coordinate is 0. The given x-intercept is (-2,0) and the y-intercept is (0,-6).

Step 2: Use the intercept form of a linear equation

The intercept form of a linear equation is given by \(\frac{x}{a} + \frac{y}{b} = 1\), where 'a' is the x-intercept and 'b' is the y-intercept.

Step 3: Plug in the intercepts

Substitute 'a' with -2 and 'b' with -6 into the intercept form equation \(\frac{x}{a} + \frac{y}{b} = 1\) to get the equation of the line.

Step 4: Write the final equation

After substitution, we get \(\frac{x}{-2} + \frac{y}{-6} = 1\). Multiply every term by -6 to clear the fractions: 3x + y = -6; it can also be written in standard form as 3x + y + 6 = 0.

Key Concepts

X-Intercept

The x-intercept of a linear equation is the point at which the graph of the line crosses the x-axis. This point indicates that the y-coordinate is zero. For instance, if you are given an x-intercept of (-2, 0), this tells you that when y is 0, x is -2.

When plotting this point on a graph or when writing an equation, you essentially hold y constant at zero and solve for the corresponding x value. The knowledge of the x-intercept is especially helpful in sketching the graph of a linear equation and in writing the equation in intercept form.

Y-Intercept

Conversely, the y-intercept is where the line crosses the y-axis, meaning the x-coordinate is zero at this point. If a line has a y-intercept of (0, -6), this indicates that when x is 0, y is -6.

This information is crucial because it provides a starting point for drawing the line on a coordinate plane and assists in formulating the equation of the line. Identifying the y-intercept is often one of the first steps to take when graphing a line or when expressing an equation in various forms, including the intercept form and the standard form.

Intercept Form of a Linear Equation

The intercept form of a linear equation is an elegant way to express lines when you know the line's intercepts. It is represented as \(\frac{x}{a} + \frac{y}{b} = 1\), where 'a' and 'b' correspond to the coordinates of the x-intercept and y-intercept, respectively.

For example, with an x-intercept at (-2, 0) and a y-intercept at (0, -6), the intercept form of the equation becomes \(\frac{x}{-2} + \frac{y}{-6} = 1\). One main advantage of this form is that it clearly displays the intercepts, making it straightforward to graph the line and also to revert back to the standard form if necessary.

Standard Form of a Linear Equation

On the other hand, the standard form of a linear equation is written as Ax + By = C, where A, B, and C are integers, and A should be non-negative. It's beneficial when you need an equation that easily allows algebraic operations such as adding or subtracting equations, or when applying the algorithm to find intercepts.

From our example, you can manipulate the intercept form \(\frac{x}{-2} + \frac{y}{-6} = 1\) through algebraic operations to find its standard form. Multiplying each term by 6 (the least common multiple of 2 and 6) gives 3x + y = -6, which can also be written as 3x + y + 6 = 0, aligning with the criterion of the standard form where coefficients are integers and A is non-negative.