Problem 84 Use the two-point form from Exer... [FREE SOLUTION]

Chapter 1: Problem 84

Use the two-point form from Exercise 83 to show that the line with intercepts $(a, 0)$ and $(0, b), a \neq 0$ and $b \neq 0,$ has the equation $$\frac{x}{a}+\frac{y}{b}=1$$

Short Answer

Expert verified

The equation of the line that passes through points $ (a, 0) $ and $ (0, b) $ is indeed $\frac{x}{a} + \frac{y}{b} = 1$.

Step by step solution

Start with the Two-Point Form

Start with the two-point equation of a straight line: $\frac{y-y_{1}}{x-x_{1}} = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$.

Substitute Correct Coordinates

Substitute the coordinates of the points ($a, 0)$ and $(0, b)$ into the equation. For the x-intercept $(a, 0)$, we have $x_{1}=a$ and $y_{1}=0$. For the y-intercept $(0, b)$, we have $x_{2}=0$ and $y_{2}=b$. This gives: $\frac{y-0}{x-a} = \frac{b-0}{0-a}$.

Simplify the Expression

Simplify the equation to $\frac{y}{x-a} = \frac{-b}{-a}$, which simplifies further into $\frac{y}{x-a} = \frac{b}{a}$ when both sides are multiplied by -1.

Cross Multiply

Next, cross multiply the equation to eliminate the fractions. This gives $ay = b(x-a)$.

Distribute b on the Right Side

Then, distribute $b$ on the right side of the equation, resulting in $ay = bx - ab$.

Rearrange the Equation into Desired Format

Rearrange the equation to the form $\frac{x}{a} + \frac{y}{b} = 1$. Rearranging gives $bx - ab = -ay$, which simplifies to $x = \frac{ab}{b} - \frac{ay}{b}$ or $x = a - \frac{ay}{b}$. Dividing both sides by a we get $\frac{x}{a} = 1 - \frac{y}{b}$ and finally, $\frac{x}{a} + \frac{y}{b} = 1$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Equation of a Line

Understanding the equation of a line starts with recognizing its importance in geometry and algebra. It鈥檚 a way to express the infinite set of points that form a straight line in a two-dimensional space.
The two-point form of a line's equation is particularly useful because it allows us to find an equation by simply knowing two points the line passes through. This form is \[\frac{y - y_1}{x - x_1} = \frac{y_2 - y_1}{x_2 - x_1}\].

It efficiently sets up a relationship that can then be manipulated into other forms, like the slope-intercept form or standard form. Knowing two points like intercepts provides a straightforward way to construct the line's equation.

This form is great for quick calculations since it directly uses available point coordinates.
It's also a stepping stone to understanding more complex linear relationships.

Intercepts

Intercepts are key concepts in understanding how a line interacts with the coordinate axes. The x-intercept occurs where the line crosses the x-axis. This means at this point, the y-coordinate is zero.
Similarly, the y-intercept is where the line crosses the y-axis, making the x-coordinate zero at that point.

In the exercise, the intercepts are given as $(a, 0)$ and $(0, b)$. These serve as input values for the two-point form of the equation.

The x-intercept, $(a, 0)$, shows where the line dips or rises across the x-axis.
The y-intercept, $(0, b)$, indicates where the line crosses vertically along the y-axis.

These intercepts help quickly sketch the line's position and behavior in the coordinate plane.

Cross Multiplication

Cross multiplication is a fundamental algebra technique that helps resolve equations involving fractions. In an expression like $\frac{y}{x-a} = \frac{b}{a}$, cross multiplication eliminates the fractions.

This step results in multiplying each side鈥檚 numerator by the opposite side鈥檚 denominator, giving us a cleaner equation like $ay = b(x-a)$.

This method is handy in simplifying complex expressions.
It ensures that all terms are properly aligned for further manipulations.

By converting the expression into a simpler form, we're able to effectively move closer to the desired equation format, free of fractions.

Rearranging Equations

Rearranging equations is the process of modifying an equation to achieve a specific form. In algebra, getting an equation into the desired form often means isolating a variable or arranging terms to simplify the expression.
In the given problem, we needed to show $\frac{x}{a} + \frac{y}{b} = 1$. This required several steps of rearranging after using cross multiplication.

Rearranging typically involves:

Distributing terms such as $b$ in $ax = b(x-a)$ to expand the equation.
Combining or shifting terms across the equal sign to structure the equation into a familiar format.

Ultimately, rearranging equations highlights the importance of understanding algebraic operations, ensuring each step logically transitions to the next while reaching the target form.

91影视

Use the two-point form from Exercise 83 to show that the line with intercepts \((a, 0)\) and \((0, b), a \neq 0\) and \(b \neq 0,\) has the equation $$\frac{x}{a}+\frac{y}{b}=1$$

Short Answer

Step by step solution

Start with the Two-Point Form

Substitute Correct Coordinates

Simplify the Expression

Cross Multiply

Distribute b on the Right Side

Rearrange the Equation into Desired Format

Key Concepts

Equation of a Line

Intercepts

Cross Multiplication

Rearranging Equations

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