Problem 32 - Understanding Elementary Algebra with Geometry

Question

Write an equation of the line satisfying the given conditions. Line has \(x\) -intercept \(-5\) and \(y\) -intercept \(-1\)

Step-by-Step Solution

Answer

The equation of the line is \(x + 5y = -5\).

Step 1: Understand the x-intercept and y-intercept

The x-intercept is the point where the line crosses the x-axis. For this problem, the x-intercept is (-5, 0). The y-intercept is the point where the line crosses the y-axis. For this problem, the y-intercept is (0, -1).

Step 2: Use the intercept form of a line's equation

The intercept form of a line's equation is \(\frac{x}{a} + \frac{y}{b} = 1\). Here, (a, 0) represents the x-intercept and (0, b) represents the y-intercept.

Step 3: Substitute the intercept values into the equation

Substitute the x-intercept \(-5\) in place of \(a\) and the y-intercept \(-1\) in place of \(b\) in the equation: \(\frac{x}{-5} + \frac{y}{-1} = 1\)

Step 4: Simplify the equation

Simplify the equation \(\frac{x}{-5} + \frac{y}{-1} = 1\) to \(\frac{-x}{5} + \frac{-y}{1} = 1\), which can be transformed to \(-\frac{x}{5} - y = 1\) by combining the terms. Multiply through by (-1) to clear the negatives: \(\frac{x}{5} + y = -1\).

Step 5: Convert to standard form (optional)

While \(\frac{x}{5} + y = -1\) is a valid form, it is often helpful to convert this to the standard form \(Ax + By = C\): Multiply every term by 5 to get: \(x + 5y = -5\)

Key Concepts

x-intercept

The **x-intercept** is a critical concept in understanding linear equations. It is the point where a line crosses the x-axis.
At this point, y is always zero. In our example, the x-intercept is (-5, 0). This tells us that when you move 5 units to the left on the x-axis, the line will touch this point.

y-intercept

Just like the x-intercept, the **y-intercept** is where the line crosses the y-axis.
Here, the x value is always zero. In this problem, the y-intercept is (0, -1). This means the line intersects the y-axis 1 unit downward from the origin.
Knowing both intercepts gives key points for sketching the line.

standard form of linear equation

The **standard form of a linear equation** is typically written as:
\[ Ax + By = C \]
Here:

• A, B, and C are constants
• x and y are the variables

Converting equations into standard form is sometimes useful for simplicity and uniformity.
In our problem, we simplified from \[ \frac{x}{5} + y = -1 \] to \[ x + 5y = -5 \]. This makes it clearer and more straightforward to understand.

intercept form of linear equation

The **intercept form** of a linear equation makes use of the line's intercepts on the axes and is written as:\[ \frac{x}{a} + \frac{y}{b} = 1 \].
In this form:

• (a, 0) is the x-intercept
• (0, b) is the y-intercept

In our example, substituting x-intercept \[-5\] and y-intercept \[-1\] gives \[ \frac{x}{-5} + \frac{y}{-1} = 1 \].
This helps in visualizing where the line touches the coordinate axes.