Problem 47 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 47-48: If a line \(l\) has nonzero \(x\) - and \(y\)-intercepts \(a\) and \(b\), respectively, then its intercept form is $$ \frac{x}{a}+\frac{y}{b}=1 . $$ Find the intercept form for the given line. $$ 4 x-2 y=6 $$

Step-by-Step Solution

Answer

The intercept form is \(\frac{2x}{3} - \frac{y}{3} = 1\).

Step 1: Identify coefficients of the equation

The given line equation is \(4x - 2y = 6\). Identify that the coefficient of \(x\) is 4, and the coefficient of \(y\) is -2.

Step 2: Convert equation to intercept form

The intercept form of a line is: \(\frac{x}{a} + \frac{y}{b} = 1\). We need to express the given equation in this form.

Step 3: Find x-intercept (a)

To find the \(x\)-intercept, set \(y = 0\) in the equation \(4x - 2y = 6\). Solve for \(x\): \(4x = 6\) yields \(x = \frac{6}{4} = \frac{3}{2}\). Thus, \(a = \frac{3}{2}\).

Step 4: Find y-intercept (b)

To find the \(y\)-intercept, set \(x = 0\) in the equation \(4x - 2y = 6\). Solve for \(y\): \(-2y = 6\) yields \(y = -3\). Thus, \(b = -3\).

Step 5: Write the intercept form

Using the values \(a = \frac{3}{2}\) and \(b = -3\), substitute into the intercept form \(\frac{x}{a} + \frac{y}{b} = 1\), we get \(\frac{x}{\frac{3}{2}} + \frac{y}{-3} = 1\). This can be simplified to \(\frac{2x}{3} - \frac{y}{3} = 1\).

Key Concepts

Linear Equations

Linear equations are expressions involving two variables, usually represented as \(x\) and \(y\), and are structured such that they plot straight lines on a graph. They have the general form \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constants. In a linear equation:

• \(A\) represents the coefficient of \(x\),
• \(B\) represents the coefficient of \(y\), and
• \(C\) is a constant term.

The equation \(4x - 2y = 6\) is an example of a linear equation. We can convert it into different forms to highlight different properties of the line. The intercept form is particularly useful for identifying where the line crosses the axes.

Finding Intercepts

The intercepts of a line are the points where it intersects the axes.

• \(x\)-intercept: The \(x\)-intercept is the point where the line meets the \(x\)-axis, which means \(y\) is zero at this point. To find it, substitute \(y = 0\) into the line equation and solve for \(x\).
• \(y\)-intercept: Conversely, the \(y\)-intercept is where the line intersects the \(y\)-axis, so \(x\) is zero here. To find it, set \(x = 0\) in the equation and solve for \(y\).

For the equation \(4x - 2y = 6\), the \(x\)-intercept (\(a\)) is \(\frac{3}{2}\), and the \(y\)-intercept (\(b\)) is \(-3\). These intercepts are essential for converting the equation into its intercept form \(\frac{x}{a} + \frac{y}{b} = 1\).

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations to extract useful information. It is a crucial step in converting an equation into its intercept form. Here’s how it’s done:

• Identify coefficients: First, recognize the coefficients of \(x\) and \(y\) in the equation: for \(4x - 2y = 6\), they are 4 and -2 respectively.
• Substitute into intercept form: Use the values of the intercepts \(a\) and \(b\) found earlier to write the line in intercept form. Substitute \(a = \frac{3}{2}\) and \(b = -3\) into \(\frac{x}{a} + \frac{y}{b} = 1\).
• Simplify: Rewrite \(\frac{x}{\frac{3}{2}} + \frac{y}{-3} = 1\) as \(\frac{2x}{3} - \frac{y}{3} = 1\), showing the line in a neatly simplified intercept form.

These steps help in communicating different features of linear equations with different forms, each highlighting unique properties. By mastering algebraic manipulation, you can switch efficiently between these forms depending on the problem at hand.