Suppose the given square matrix is called A. To find the inverse of any matrix, we write the matrix in a larger matrix along side an identity matrix of the same size,
Now use row operations to rewrite this matrix so that the identity appears on the left side. The inverse of the original matrix will be on the right side of the transformed matrix,
In this FAQ, I’ll demonstrate how we can use y = mx + b to find equations of lines. No need to memorize other equations of lines…it is easier to focus on the data given to us an use it to find m and b.
Let’s look at the most basic example that illustrates this process.
a. Find the equation of a line through the point (4, -6) with slope – 3/4.
Start by substituting m = -3/4 in the slope-intercept form to yield
y = – 3/4 x + b
Now substitute the point into the line by setting x = 4 and y = -6. This leads to
Using this value for b in the slope-intercept form above give the line y = –3/4x -3. The answer to the problem is the equation of the line with the appropriate values for m and b.
Now let’s complicate matters a bit by finding the equation of a line passing through two points.
b. Find a line through the points (-1, 3) and (2, 6).
The slope through the points is
Substitute the slope into the slope-intercept form to give
y = x + b
Now take one of the points and substitute it into this equation. Using the ordered pair (2, 6), we can solve for b:
This gives us the equation y = x + 4.
In both of these examples we applied the same strategy of putting in the slope m and then solving for the intercept b.
Now let’s look at a problem whose wording might throw you off.
c. Find a line with x-intercept of -5 and a y-intercept of 4.
Don’t let the fact that they talk about intercepts throw you off the strategy. These intercepts can be written as ordered pairs (-5, 0) and (0, 4). The slope between these points is
This leads to the line
y = 4/5x + b
Since the y-intercept is 4, we can substitute it into this line for b to give y = 4/5x + 4 .
This problem is even easier since the y intercept was given to us. In each case we can start from y = mx + band then find the value of m and b.
In another MathFAQ, I examined how we can find the equation of a line from two data points. In this post I want to look at a closely related problem where we find the equation of the line from a rate.
Problem Assume the growth of the population of Del Webb’s Sun City Hilton Head community was linear from 1996 to 2000, with a population of 198 in 1996 and a rate of growth of 705 people per year.
a. Write an equation for the population P of the community where x is the number of years after 1990.
Solution The population of 198 in 1996 corresponds to the point (6, 198) since the variable x corresponds to years after 1990. We’ll write the slope-intercept form of the line, P = mx + b, and substitute m into the equation. The rate of growth, 705 people per year, is the slope of the function. Therefore, the line describing the population is
P = 705x + b
To find the value of b, we need to substitute x = 6 and P = 198:
198 = 705(6) + b
-4032 = b
This gives us the equation,
P = 705x – 4032
b. Use the function to estimate the population in 2002.
Solution The year 2002 corresponds to x = 12. Substitute this value into the function to yield
P = 705(12) – 4032 = 4428
The population in 2002 will be 4428 people.
c. In what year will the population reach 10,000?
Solution In this part, set P = 10,000 and solve for x.
10000 = 705x – 4032
14032 = 705x
14032/705 = x
19.9 ≈ x
This corresponds to 1990 + 19.9 = 2009.9. So in the year 2009, the population will reach 10000. Since we are asking in what year, we DO NOT round up on the answer.
Problem For the parabola y = 2x2 + 3x – 2, locate the points below.
a. The y-intercept.
Solution At the y-intercept, the x value is zero. This means that we need to set x = 0 in the equation:
y = 2(02) + 3(0) – 2 = -2
Putting this together, the y-intercept is at (0, -2).
b. The vertex.
Solution The vertex is located using the formula where the values of a, b, and c come from the equation. In this case, a = 2, b = 3, and c = -2. This gives an x value on the intercept of
To find the corresponding y value, put this value into the equation,
This means the vertex is at (-3/4, –25/8).
c. The x-intercepts.
Solution At the x-intercepts, the y value is zero. Putting this into the equation yields
0 = 2x2 + 3x – 2
This equation is solved with the quadratic formula,
Put the values from the equation (a = 2, b = 3, and c = -2),
The x intercepts are at (-2, 0) and (1/2, 0).
All of these points are shown in the graph of the parabola below.
Below are some problems that students solved on the board in previous semesters. All of them start from the slope-intercept form of a line and require you to find the slope m from two points and then solve for b. Click on the pictures to see a larger version.
Problem The percent p of adults who smoke cigarettes can be modeled by a linear equations p = mt + b, where t is the number of years after 1960. If two points on the graph of this function are (25, 30.7) and (50, 18.1), write the linear equation of this application.
Solution
Problem The number of women in the workforce, based on data and projections from 1950 to 2050, can be modeled by a linear equation y = mx + b. The number was 18.4 million in 1950 and is projected to be 81.6 million in 2030. Let x represent the number of years after 1950 and y be the number of women in the workforce in millions.
a. What is the slope of the line through (0, 18.4) and (80,81.6)?
b. What is the average rate of change in the number of women in the workforce during this time period?
c. Use the slope from part a and the number of million of women in the workforce in 1950 to write the equation of the line.
In 
