Problem 71 The intercept form of a line is ... [FREE SOLUTION]

91影视

College Algebra with Modeling and Visualization

Gary K. Rockswold

$Math Studyset 91影视 Explanations$ Math

4 Edition

Chapter 2: Problem 71

The intercept form of a line is $\frac{x}{a}+\frac{y}{b}=1$ Determine the $x$ -and y-intercepts on the graph of the equation. Draw a conclusion about what the constants a and b represent in this form. $$ \frac{2 x}{3}+\frac{4 y}{5}=1 $$

Short Answer

Expert verified

The x-intercept is $(\frac{3}{2}, 0)$ and the y-intercept is $(0, \frac{5}{4})$. Constants $a$ and $b$ are the x- and y-intercepts respectively.

Step by step solution

Identify Intercept Form

The line's equation given is $\frac{2x}{3} + \frac{4y}{5} = 1$. This matches the general intercept form of a line $\frac{x}{a} + \frac{y}{b} = 1$. Here, $a$ and $b$ are the coefficients of $x$ and $y$ when expressed as fractions with the denominator as the constant 1 on the right-hand side.

Determine x-intercept

To find the $x$-intercept, set $y = 0$ in the equation $\frac{2x}{3} + \frac{4y}{5} = 1$. Substituting, we get $\frac{2x}{3} = 1$. Solving for $x$ gives: \[ x = \frac{3}{2} \] Thus, the $x$-intercept is $(\frac{3}{2}, 0)$.

Determine y-intercept

To find the $y$-intercept, set $x = 0$ in the equation $\frac{2x}{3} + \frac{4y}{5} = 1$. Substituting, we get $\frac{4y}{5} = 1$. Solving for $y$ gives: \[ y = \frac{5}{4} \] Thus, the $y$-intercept is $(0, \frac{5}{4})$.

Conclusion about a and b

In the intercept form $\frac{x}{a} + \frac{y}{b} = 1$, $a$ represents the $x$-intercept and $b$ represents the $y$-intercept of the line. From the given line equation, the coefficients $\frac{3}{2}$ and $\frac{5}{4}$ show that $a = \frac{3}{2}$ and $b = \frac{5}{4}$, which are indeed the intercepts.

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Key Concepts

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

Understanding X-Intercept

The x-intercept of a line is a crucial concept when analyzing linear equations. To find the x-intercept, you look for the point where the graph of the equation crosses the x-axis. At this point, the y-coordinate will always be zero because it's right on the axis itself.
For example, if you are given the equation $ \frac{2x}{3} + \frac{4y}{5} = 1 $, you can find the x-intercept by setting $ y = 0 $.

Insert the value into the equation: $ \frac{2x}{3} + \frac{4(0)}{5} = 1 $.

This simplifies to $ \frac{2x}{3} = 1 $.

Solving for $ x $, multiply both sides by 3 to get $ 2x = 3 $, and then divide by 2 to find $ x = \frac{3}{2} $.

Therefore, the x-intercept is the point $ (\frac{3}{2}, 0) $. This information helps identify where the line touches the x-axis and is essential for graphing linear equations.

Identifying Y-Intercept

The y-intercept is another key feature when looking at linear equations. It indicates where the graph of the equation intersects the y-axis. At this point, the x-coordinate will always be zero.
To find the y-intercept in the given equation $ \frac{2x}{3} + \frac{4y}{5} = 1 $, set $ x = 0 $.

Substitute the value into the equation: $ \frac{2(0)}{3} + \frac{4y}{5} = 1 $.

This simplifies to $ \frac{4y}{5} = 1 $.

Solving for $ y $, multiply both sides by 5 to get $ 4y = 5 $, and then divide by 4 to obtain $ y = \frac{5}{4} $.

Thus, the y-intercept is $ (0, \frac{5}{4}) $. Knowing this point is crucial for understanding how the line behaves in relation to the y-axis, further illustrating the relationship between x and y.

Exploring Linear Equations

Linear equations are fundamental in mathematics, representing straight lines when graphed. The intercept form of a linear equation, $ \frac{x}{a} + \frac{y}{b} = 1 $, offers a clear way to find both intercepts immediately.
Here are some key features of linear equations in intercept form:

Simplicity: The equation is straightforward because it directly reveals both the x-intercept $ a $ and y-intercept $ b $. Each intercept can be found by setting the other variable to zero, simplifying the computational process.

Purpose: Using this form helps to quickly determine where a line crosses the axes, which is essential for graphing and interpreting data in various practical applications.

Visual Understanding: By knowing the intercepts, one gains insights into the slope and direction of the line without complex calculations.

The form facilitates a deeper intuition for how changes in intercepts affect the graph, thereby aiding in more efficient problem solving.

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91影视

The intercept form of a line is \(\frac{x}{a}+\frac{y}{b}=1\) Determine the \(x\) -and y-intercepts on the graph of the equation. Draw a conclusion about what the constants a and b represent in this form. $$ \frac{2 x}{3}+\frac{4 y}{5}=1 $$

Short Answer

Step by step solution

Identify Intercept Form

Determine x-intercept

Determine y-intercept

Conclusion about a and b

Key Concepts

Understanding X-Intercept

Identifying Y-Intercept

Exploring Linear Equations

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