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Chapter 2: Problem 37

Write an equation of the line satisfying the following conditions. Write the equation in the form \(y=\mathrm{mx}+b\). It passes through the point (3,10) and has slope \(=2\).

Short Answer

Expert verified
The equation of the line is \(y = 2x + 4\).

Step by step solution

01

Understand the formula

The formula of a straight line is \(y=mx+b\), where \(m\) is the slope and \(b\) is the y-intercept.
02

Substitute the slope

The problem gives a slope of 2. So, place it into the equation: \(y=2x+b\).
03

Substitute the point

We know the line passes through the point (3,10). Substitute these values into the equation: \(10 = 2*3 + b\).
04

Solve for the y-intercept

Solve the equation from step 3 for \(b\). This gives \(b = 10 - 6 = 4\).

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Key Concepts

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

Slope
The slope of a line measures how steep the line is. It is a critical component when you are working with linear equations. Slope can be seen as the 'rise over run', which is also how much the line rises vertically for a unit increase in horizontal distance.
Understanding the concept of slope can help you determine how changes in the x-variable affect the y-variable.
  • If the slope is positive, the line rises as you move from left to right.
  • If the slope is negative, the line falls as you move from left to right.
  • A zero slope means the line is perfectly horizontal, making it level with the x-axis.
  • An undefined slope is characteristic of a vertical line.
The formula for calculating slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \( \frac{y_2 - y_1}{x_2 - x_1} \). This formula tells us how much y changes for each change in x.
Y-intercept
The y-intercept is the point where the line crosses the y-axis. Knowing the y-intercept is helpful because it gives you a point on the line with the x-value of 0. This intercept is usually denoted by the letter \(b\) in the linear equation \(y = mx + b\).
For example, in the equation \(y = 2x + 4\), the y-intercept is 4, meaning when \(x = 0\), \(y = 4\). It gives you important information about where your line starts on the graph.
  • If \(b\) is positive, the line crosses the y-axis above the origin.
  • If \(b\) is negative, the line crosses below the origin.
  • If \(b = 0\), the line passes through the origin itself.
In applications, y-intercepts can represent starting values in word problems or real-life scenarios.
Equation of a line
The equation of a line is a mathematical formula that represents all the points lying on that line. The most common form of this equation is the slope-intercept form, expressed as \(y = mx + b\). Here, \(m\) is the slope and \(b\) is the y-intercept.
To construct the equation of a line, you need to know the slope and at least one point the line passes through.
  • Slope, \(m\), indicates how steep the line is.
  • Y-intercept, \(b\), shows where the line crosses the y-axis.
To find the equation of a line given a point and a slope, substitute the slope into the equation and solve for \(b\) using the coordinates of the point. Once you have \(b\), substitute it back into the equation to get the full equation of the line.
Point-slope form
Point-slope form is another way to write the equation of a line. It's often more useful when you have a point on the line and the slope, but don't initially have the y-intercept. The equation looks like \(y - y_1 = m(x - x_1)\).
  • \((x_1, y_1)\) is a known point on the line.
  • \(m\) is the slope.
Using the point-slope form makes finding the equation of a line straightforward if you don't yet know the y-intercept. You can rearrange it into the standard slope-intercept form by solving for \(y\). For example, if a line passes through the point (3, 10) and has a slope of 2, the point-slope form would be \(y - 10 = 2(x - 3)\). After solving for \(y\), you get the slope-intercept form \(y = 2x + 4\), which makes it easy to graph or analyze further.

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Most popular questions from this chapter

In 1955 , an average new Chevrolet sold for $$\$ 2,400$$, and a similar Chevrolet sold for $$\$ 15,000$$ in 1995 . Assuming a linear relationship, write an equation that will give the price of an average Chevrolet in any year. Use this equation to predict the price of an average Chevrolet in the year 2010 .

Find the slope of the line passing through the following pair of points. (4,3) and (-1,3)

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