Introduction
In the previous lesson (problem 38), we were asked to prove that the three medians of the triangle $A\left( { - 3,13} \right)$, $B\left( {9,1} \right)$ and $C\left( {-9,1} \right)$ all contained the point $\left( { - 1,5} \right)$, after first finding the equations of the medians. But what if you were not given the point $\left( { - 1,5} \right)$? How would you discover that point from the equations of the medians?
More generally, given the equations of two lines, how could you find the point where those lines intersect — the solutions that the two equations share? The hunt for solutions that equations share is a useful and sometimes exciting quest, and can reveal interesting relationships between the equations.
One could answer the question of finding a common solution by guessing, but there are times when guessing is extremely difficult. In this lesson we will explore algebraic methods of exploring common solutions to a set of linear equations — called a linear system of equations. At the same time we will use visual approaches to promote better understanding of the relationships between the variables, and the nature and existence of common solutions.
Development
Recall that the equations of the medians in the previous lesson (problem 38) were:
- $ \large{\frac{1}{2}x + \frac{11}{2}}$
- $ \large{y = - 4x + 1}$
- $ \large{y = \frac{{ - 2}}{5}x + \frac{{23}}{5}}$
For the time being let’s just work with the first two equations and try to find a particular ordered pair $\left( {x,y} \right)$ which satisfies both — values of $x$ and $y$ which make both of them work.
Since the second equation claims that $y$ is equal to $- 4x + 1$, it seems okay to replace $y$ by $- 4x + 1$ in the first equation.
What does the equation become when $y$ is replaced by $- 4x + 1$? Now solve this equation for $x$.
Substitute the value of $x$ you got in problem 1 in the first equation to get the value of $y$.
Use your results from problems 1 and 2 to show that the medians all intersect in (contain) a single point.
The form of the median equations was particularly convenient to make the replacement/substitution for $y$. Could you do a similar replacement if the equations were written in a different form? Fortunately most linear equations can be written in a form similar to one of the median equations. Take a look at the following problem.
Find the point where the lines given by the equations $5x + y = 1$ and $x - 5y = 8$ intersect.
Note that from the first equation we get $y = 1 - 5x$, so replacing $y$ by $1 - 5x$ in the second equation should be fair game. Replace $y$ by $1 - 5x$ in the second equation and solve for $x$.
Use the value of $x$ you got in part a and the fact that to find the value of $y$. Check to be sure that the values you got for $x$ and $y$ satisfy both of the equations.
Solve the pair of equations in problem 4 by starting with the second equation. (Express $x$ in terms of $y$ and substitute for $x$ in the first equation.)
By the way, the method we used to find the solution in problems 1 through 5 is called the substitution method.
Use the method of substitution to solve the following systems of equations:
$x + 4y = - 29$
$3x - 2y = 11$
$3x - 5y = 4$
$5x - y = 3$
Practice
Use the method of substitution to solve the following systems.
$x + y = 20$
$x - y = 10$
$x + y = 21$
$y = 2x$
$2p + q = 0$
$4p - q = 3$
$4r - 3s = -11$
$s + 2r = 2$
Matt claims that he discovered that there is one particular ordered pair that would satisfy all three equations because when he graphed the lines they all met in a single point. Gabbi drew the graphs herself and said that she wasn’t quite sure. Matt could be right but she thought it was too close to call. Check Matt’s claim algebraically.
- $4x + 2y = - 1$
- $5x - $y$ = 4$
- $17x + 5y = 1$
Problems
Solve the system of equations below:
$5x + 3y = - 2$
$7x - 5y = 11$
Is the ordered triple
$\left( { -2,3,\frac{1}{2}} \right)$ a solution to the following system of
equations? Explain your response. If it is not a
solution, change one of the equations so that the given
ordered triple is a solution.
$3y - 2x + 4z = 15$
$x - 2z + 3y = 6$
$11 + 5x = 6z - y$
Determine whether the following three equations share a
common solution.
$3x + 2y = 4$
$5x - 2y = 0$
$4x + 3y = 6$
Create a system of two different equations in two variables that has as a solution the ordered pair $\left( { - 1,3} \right)$.
Jocelyn tells Joey that she has found the intersection point of the two lines $y = - 8x + 33$ and $y = 2x - 8$, and that it is close to the $x$-axis. Without solving the system exactly, can you confirm or debunk Jocelyn’s claim quickly?
After the class had completed problem 9 using the substitution method, Sophie (of course) proclaimed that this method was a waste of time and that Jeff had taught her an easier way, and that it was a lot more fun.
“Simply multiply both sides of the first equation by 5, multiply both sides of the second equation by 3, and then add your two new equations together. At this point you are pretty much done.”
Check out Sophie’s method.
“Ah”, declares Sampson, always trying to upstage Sophie, “
but we could easily have multiplied the first equation by 7,
the second equation by -5, and we’d get pretty much the same
thing.”
Check out Sampson’s approach.
The method Sampson and Sophie used in problems 14 and 15 to solve the system of equations in problem 9 is called the method of elimination.
Use the method of elimination to solve the following systems.
$2x - 5y = 9$
$7x + 3y = 11$
$2x + 3y = - 9$
$3x - 2y = 19$
$7p + 3q = - 1$
$5p + 5q = - 5$
$3r + 7s = 41$
$4r + 8s = 48$
Solve the following system (in your head, if you
can!):
$5,000,000,000,000x$
$\quad + 6,000,000,000,000y = 28,000,000,000,000$
$2 \cdot {10^{ - 5}}x + 3 \cdot {10^{ - 5}}y = 13 \cdot {10^{
- 5}}$
For the next few problems you will find a carefully drawn diagram to be quite useful.
When Jamie tried to solve the following systems of
equations she encountered all kinds of strange difficulties.
Try to figure out why.
$6x - 15y = 4$
$-4x + 10y = -8$
Solve the following systems of equations. What is going on
here?
$6x - 10y = -1$
$-9x + 15y = \frac{3}{2}$
Determine the area of the triangle formed by the following
lines.
$y = 4x - 14$
$4y = -x - 22$
$6y = 7x - 16$
You have an aquarium with some one-tailed fish and some two-tailed fish. Total, there are 102 eyes and 62 tails. How many of each kind are there?
Here are two functions: € takes a number, multiplies it by 3, and subtracts the answer from 8; ∏ takes a number, divides it by 4, then adds 0.125, and finally multiplies the result by 3. Is there any input for which these two functions have the same output?
Given $A(0,0)$ and $B(1,1)$, is there a point $C(x,y)$ such that the slope of the line $AC$ is 3 and the slope of the line $BC$ is 2?
Use your calculator to solve the following systems of
equations within two decimal place accuracy. (Hint: First
write both equations in the form $y = mx + b$.)
$117x - 144y = 135$
$258x + 336y = - 138$
A sum of \$20,000 is to be invested in two funds for a year, split among the funds. One fund, the less risky one, is expected to pay 6.3% interest for the year. The second fund is expected to pay an interest of 7.6% for the year. If one wants to earn at least \$1350 in interest at the end of the year, what is the most that can be invested at 6.3%?
(You could represent the amount invested at 6.3% by some variable, and the amount invested at 7.6% by another variable. Then try to set up a system of two equations. Or you may try to guess at a solution and use that guess to refine your thinking.)
A salesperson, Emma, is offered the following salary plans. In one plan, she gets a straight commission of 5.9% on all sales. In the other plan she gets a commission of 3.1% on sales, and a salary of \$250 per week. How much in sales would Emma have to make in a week to make the straight commission a better offer?
Kristin challenged her mother to a race, but demanded a head start of 20 meters. Kristin’s running speed is 5.5 meters per second, while her mother’s running speed is 6 meters per second. How many seconds after the start of the race will Kristin’s mother catch up with her?
Christina’s company processes a roll of film for 30 cents per print, plus a 66 cents developing charge. Nate’s company processes a roll for 28 cents per print, plus a \$2.10 developing charge. After how many prints will the cost at Christina’s company exceed that of Nate’s?
The sum of the digits of a two-digit number is one more than three times the units digit. When the digits of the two-digit number are reversed the number is reduced by 36. First write two equations representing the information, then solve them to find the number.
Gina has dimes and quarters (and no other coins) in her pocket. The total value of the 13 coins is \$2.65. First write two equations representing the information, then solve them to find the number of quarters Gina has.
You previously saw that the medians of any triangle intersected at a single point. Do the altitudes of a triangle do the same? Test your answer on the triangle, $A\left( {3,3} \right)$, $B\left( {9,6} \right)$ and $C\left( {6,9} \right)$.
A two-digit number is equal to four times the sum of its digits. The tens digit is 3 less than the units digit. What is the number?
The Matrix games store sells two types of ping-pong sets: A standard set consisting of two paddles and one ball, and a tournament set consisting of four paddles and six balls. The store receives a bulk shipment of 160 paddles and 180 balls. How many of each type of ping-pong set can be made from this shipment?
If you added up Fred and Wilma’s ages 15 years ago, you’d get Fred’s current age. What are the possibilities for Fred and Wilma’s ages now?
You mail 100 oz of letters. For regular mail, you pay 39 cents for the first ounce and 24 cents per additional ounce. For premium mail, you pay 31 cents per ounce. In total, you paid \$26.11. How many ounces did you send, in each kind of mail?
It seems that the following lines might form a rhombus.
What do you think?
$3y - 2x = 5$
$y = 2x-1$
$3y=2x+13$
$y-2x=3$
Exploring in Depth
Try to find the coordinates of the points where:
the line $y = - 3x$ intersects the graph of the equation $x^2 + y^2 = 100$.
the line $y = - 10$ intersects the graph of the equation $x^2 + y^2 = 100$.
The slope of the line segment from $\left( {6,1} \right)$ to $\left( {x,y} \right)$ is 7, and the slope of the line segment from $\left( {5,6} \right)$ to $\left( {x,y} \right)$ is 3. What is $\left( {x,y} \right)$?
The graphs of the following equations intersect at the
point $(\frac{1}{2}, \frac{3}{2})$. Find the values of $a$ and $b$.
$ax+by=3$
$bx-ay=-1$
Find a and $b$ if $\left( {10,b} \right)$ solves the
following equations.
$ax+3y=52$
$5x-5y=2a$
Paul and John each have some coins. At first, Paul has 4 times as many coins as John. Then Paul gives John 6 coins. Now Paul has 2.5 times as many coins as John. How many coins did each of them have originally?
Don’t use a calculator for this problem.
Find $( - 5 + 8)[2( - 1 + 3) - 4( - 3 - 2)]$
Find an equation of the line that goes through the points $(-6, -23)$ and $(8, -2)$.
Factor $10{x^2}y + 2xy$
Solve for x: $\frac{5}{{x - 1}} = 3$
What are the possibilities for $x$ if $|x - 2| < 4$?
Construct your own problem similar to the original problem in the introduction to this lesson, but not just a translation of that triangle, in which the coordinates of the point of intersection of the medians are both integers.
You are filling an aquarium with plankton and algae. A scoop of plankton weighs .86 lbs, and uses .14 gallons of oxygen in a day. A scoop of algae weighs .71 lbs, and produces .21 gallons of oxygen in a day. You want the total weight to be 12 lbs, and you want the oxygen to balance out perfectly. How many scoops of each should you buy?
If Bill had four more books, he’d have half as many as Tony. If Tony had one more book, he’d have as many as Bill and Mimi combined. Mimi has $\frac{3}{4}$ as many books as Tony. How many books does each person have?
Find all solutions to the system of
inequalities below.
$x - y \leq 9$
$2x + y \geq 3$
Write two linear equations that have the solution $(\sqrt {11,} \;\frac{{ - 1}}{3})$.
Solve the following system. Think about how you solve
linear equations if you’re stuck!
$x^2 + y^2 = 100$
$x^2 - y^2 = 64$
Solve the following system.
$\frac{x}{y} = 12$
$xy = 75$
Solve the following system.
$y=\frac{1}{x}$
$y=x^2$
Find $a$ and $b$ if $(25,16)$ solves the
following system.
$y-b=\sqrt{x+a}$
$y=\sqrt{x}+a$
Solve the following system of equations for $x$, $y$ and
$w$.
$-2x+3y+4w=6$
$x+3y-2w=6$
$x+y-6w=0$