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Chapter 7: Problem 17

For the following problems, find the equation of the line using the information provided. Write the equation in slope-intercept form. slope \(=7, \quad\) passesthrough (0,0) .

Short Answer

Expert verified
Answer: The equation of the line is y = 7x.

Step by step solution

01

Identify slope and point coordinates

The given slope is \(7\) and the point through which the line passes is \((0,0)\). This means our line has a slope of \(7\) and passes through the origin \((0,0)\), which is also the y-intercept.
02

Plug the values into the slope-intercept equation

We know the slope-intercept form of the equation is \(y = mx + b\). We have the slope \(m = 7\) and the point \((x,y) = (0,0)\). To find the y-intercept (b), plug these values into the equation: \(0 = 7 \cdot 0 + b\)
03

Solve for the y-intercept (b)

Simplify the equation to solve for b: \(0 = 0 + b\) Since the only term left on the right side of the equation is b, it is equal to 0: \(b = 0\)
04

Write the final equation

Now that we have the slope (m) and the y-intercept (b), we can write the equation of the line in slope-intercept form: \(y = 7x + 0\) However, since adding 0 does not change the value of the expression, we can simply write: \(y = 7x\) The equation of this line in slope-intercept form is \(y = 7x\).

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Key Concepts

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

Linear Equations
A linear equation represents a straight line when plotted on a coordinate plane. These equations are generally expressed in the form of Ax + By + C = 0, where A, B, and C are constants. However, in different contexts, like the slope-intercept form, it can be written as y = mx + b. This form is particularly useful as it directly reveals important characteristics of the line, such as the slope and the y-intercept.

Key aspects of linear equations include:
  • Each equation defines a relationship between x and y.
  • The solution to a linear equation represents a set of points that creates a straight line.
  • They have constant rates of change, represented by the slope.
Being able to understand and manipulate linear equations is foundational in algebra and essential for understanding more complex mathematical concepts.
Slope
The slope of a line is a measure of its steepness and direction. It's often represented by the letter 'm' in the slope-intercept form of a linear equation, y = mx + b. The slope is calculated as the ratio of the change in the y-values to the change in the x-values between any two points on the line. Mathematically, it is expressed as:\[m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}\]
  • A positive slope indicates a line ascending from left to right.
  • A negative slope points to a line descending from left to right.
  • A slope of zero represents a perfectly horizontal line, while an undefined slope is vertical.
Understanding slope is crucial not just in math but in various real-world applications, such as determining speed, growth rates, and economic trends.
Y-intercept
The y-intercept of a line refers to the point where the line crosses the y-axis. In the context of the slope-intercept form, the y-intercept is represented by 'b'. This value is essential because it provides a starting point for graphing a linear equation. For example, in the equation y = 7x + 0, the y-intercept is 0. This means the line passes through the origin (0,0).
  • The y-intercept tells us the value of y when x equals 0.
  • It helps in fully understanding the positioning of the line on a coordinate plane.
Knowing how to find and interpret the y-intercept allows students to quickly visualize where a line will start on the y-axis and is vital for graphing lines by hand or using technological tools.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, involves the study of geometry using a coordinate system. This branch of mathematics helps us understand geometric properties of shapes and lines with the help of algebra. In coordinate geometry:
  • Points are represented by coordinates (x, y).
  • Lines are expressed using equations like the slope-intercept form.
  • It combines algebraic principles with geometric visualization.
Through coordinate geometry, we can calculate distances, midpoints, and slopes, and graph lines and shapes on a plane. It's a powerful tool for solving real-world geometric problems, such as finding the shortest path between two points or visualizing data trends. Knowing how to use coordinate geometry can deeply enhance one's ability to analyze and solve geometric challenges.

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Most popular questions from this chapter

Determine the slope and \(y\) -intercept of the lines. $$ y=x $$

For the following problems, find the slope of the line through the pairs of points. Round to two decimal places. $$ (33.1,8.9),(42.7,-1.06) $$

Determine the slope and \(y\) -intercept of the lines. $$ y=9 x-1 $$

Simplify \(\left(x^{2} y^{3} w^{4}\right)^{0}\)

For the following problems, find the slope of the line through the pairs of points. Round to two decimal places. $$ (5.56,9.37),(2.16,4.90) $$
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