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Chapter 1: Problem 63

Find an equation of the line that satisfies the given conditions. \(x\) -intercept \(4 ; y\) -intercept -3

Short Answer

Expert verified
The equation of the line is \( y = \frac{3}{4}x - 3 \).

Step by step solution

01

Recognize Intercepts

An intercept is where a graph crosses the axis. The x-intercept is where the graph crosses the x-axis, and here it's given as 4, which means the point (4, 0) is on the line. The y-intercept is where the graph crosses the y-axis, and here it is given as -3, meaning the point (0, -3) is on the line.
02

Find Slope of the Line

Use the formula for slope, which is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Substitute the intercept points \((4, 0)\) and \((0, -3)\) into the formula:\[ m = \frac{0 - (-3)}{4 - 0} = \frac{3}{4} \]. Thus, the slope of the line is \( \frac{3}{4} \).
03

Write Equation in Point-Slope Form

Using the slope \( m = \frac{3}{4} \) and one of the points, let's use \((0, -3)\), apply the point-slope formula to write the equation: \( y - y_1 = m(x - x_1) \). Plugging in the values gives: \[ y - (-3) = \frac{3}{4}(x - 0) \]. Simplifying, \[ y + 3 = \frac{3}{4}x \].
04

Convert to Slope-Intercept Form

To put the equation into slope-intercept form \( y = mx + b \), solve for \( y \):\[ y = \frac{3}{4}x - 3 \]. This shows that the equation of the line is \( y = \frac{3}{4}x - 3 \), with slope \( \frac{3}{4} \) and y-intercept \(-3\).

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Key Concepts

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

x-intercept
The "x-intercept" of a line in a coordinate plane is the point where the line crosses the x-axis. In mathematical terms, this is where the y value is zero. For example, if a line crosses the x-axis at the point (4, 0), the x-intercept is 4.

Identifying the x-intercept is crucial because it provides one of the coordinates needed to determine the line's equation.
  • It helps in calculating the slope of the line when you have another point on the line.
  • Each x-intercept will have its own significance based on the context of a problem.
Knowing how to find the x-intercept can also help check the correctness of a graph or an equation.
y-intercept
The "y-intercept" is where the line crosses the y-axis. This point always has an x-coordinate of 0. If the line crosses the y-axis at (0, -3), then the y-intercept is -3. This concept is just as important as the x-intercept, providing the necessary information to formulate a line's equation.

Here are some key points to remember:
  • The y-intercept is often represented by the constant "b" in the slope-intercept form of a line.
  • It represents the point on the graph where the line starts on the y-axis.
The y-intercept is particularly useful in real-world scenarios, as it represents starting values when dealing with equations of linear models.
slope-intercept form
The "slope-intercept form" of a linear equation is expressed as: \[ y = mx + b \]where "m" is the slope, and "b" is the y-intercept. This form is incredibly useful for quickly identifying the slope of the line and where it crosses the y-axis.

Understanding slope-intercept form allows for:
  • Easy graphing with the slope and y-intercept guiding the direction and position of the line.
  • Efficient translation of real-life situations into mathematical expressions.
By expressing the equation as such, you simplify calculations for any given x-value.
point-slope form
The "point-slope form" is another way to express the equation of a line. It is especially useful when you know one point on the line and the slope. The formula is presented as: \[ y - y_1 = m(x - x_1) \]where (x_1, y_1) is the known point and "m" is the slope.

Some benefits of using the point-slope form include:
  • Flexibility in identifying an equation when different points are known.
  • Easy derivation to arrive at the slope-intercept form.
Understanding this form allows you to adapt to varied problems effectively, especially in geometry and algebra.

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Most popular questions from this chapter

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