12. Algebraic Manipulations and Equations

12.10%

System of Equations Basics

In this section you will be able to solve an equation by doing the same thing to both sides, and solve quadratics by factoring or the formula.

Three mystery boxes and 4 weights balance one mystery box and 10 weights. All the boxes weigh the same. How heavy is one box? A balance scale can tell you!

An equation is a balance scale. Whatever you do to one pan, do to the other pan, and it stays balanced.

Take the same things off both sides until one box sits alone. That is solving for $x$ .

With two unknowns, you need two equations. You can substitute one into the other, or eliminate a letter by adding or subtracting the equations.

Look:

Quadratics like $x^2 - 5x + 4 = 0$ need a new trick. Factor into $(x - 1)(x - 4) = 0$ : one bracket must be zero.

Exploreplay first, formulas later

Try this: Use the Try x stepper. Which value of x makes the scale balance?
Try this: Press Take x off both pans, then Take 4 off both pans. Does the scale stay balanced? Why?
Try this: Press Split both pans into 2 equal parts. What does one box weigh?
Try this: Turn on Show the formula. Each move is one line of algebra.

Name it

Concept 12.1.1 · Factoring a Quadratic

If we have a quadratic of the form $x^2 + bx + c$ , it can be factored into $(x + p)(x + q)$ such that $p + q = b$ and $pq = c$ . We can generally do this by guess and check.

Then, we just solve the linear equations of the form $x + r = 0$ .

Concept 12.1.2 · Vieta's Formulas

If we have a quadratic of the form $x^2 + bx + c$ with 2 roots $r$ and $s$ , then $r + s = -b$ and $rs = c$ . Sometimes, you may not need to calculate the roots and just find expression in terms of the roots.

Theorem 12.1.3 · Quadratic Formula

The solutions to the quadratic equation

ax^2 + bx + c = 0

are

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.

Remark 12.1.5 · Solving 2 variable equations

The most common ways of solving equations are substitution and elimination.

Linear Equations Basics

There is an easier way to solve 2 variable equations using the diagonal product method:

Diagonal Product Method to Solve 2 Variable Equations

Try it

Problem 12.4.1

AMC 8

Shauna takes five tests, each worth a maximum of 100 points. Her scores on the first three tests are 76, 94, and 87. In order to average 81 for all five tests, what is the lowest score she could earn on one of the other two tests?

Problem 12.4.4

AMC 8

In a mathematics contest with ten problems, a student gains 5 points for a correct answer and loses 2 points for an incorrect answer. If Olivia answered every problem and her score was 29, how many correct answers did she have?

Problem 12.4.9

AMC 8

Suppose that $a * b$ means $3a - b$ . What is the value of $x$ if

2 * (5 * x) = 1

Contest cornerreal contest problems

Problem 12.4.10

AMC 10

In an after-school program for juniors and seniors there is a debate team with an equal number of students from each class on the team. Among the 28 students on the program, $25\%$ of the juniors and $10\%$ of the seniors are on the debate team. How many juniors are in the program?

Problem 12.4.20(additional)

AMC 10

Ana and Bonita were born on the same date in different years, n years apart. Last year Ana was 5 times as old as Bonita. This year Ana's age is the square of Bonita's age. What is n?

Problem 12.4.21(additional)

AMC 8

On the last day of school, Mrs. Awesome gave jelly beans to her class. She gave each boy as many jelly beans as there were boys in the class. She gave each girl as many jelly beans as there were girls in the class. She brought 400 jelly beans, and when she finished, she had six jelly beans left. There were two more boys than girls in her class. How many students were in her class?

Problem 12.4.25(additional)

MATHCOUNTS

If

\frac{3}{2 - x} + \frac{2}{y - 3} = 1

and

\frac{2}{2 - x} + \frac{8}{y - 3} = 2

find the value of $x$ .

Checkpoint for section 12.1

Get 2 of 3 right to master this section. One try each, no hints.

12.20%

Advanced Equation Solving Techniques

In this section you will be able to choose a smart method for a system: symmetry, adding equations, factoring or a graph.

Contest systems often look scary: three equations, three unknowns. But they usually ask for something like $x + y + z$ , not each letter. Look for the shortcut!

Before you start solving, look at the shape of the equations. Are they symmetric? Do they share a factor? Can you add them all up?

Often you never find $x$ , $y$ and $z$ one by one. You find the expression the problem asks for directly.

Look:

Two big helpers: $x^2 - y^2 = (x - y)(x + y)$ and $(x + y)^2 = x^2 + 2xy + y^2$ .

Exploreplay first, formulas later

Line 1

2

x +

3

y =

12

Line 2

1

x +

-1

y =

1

One solution: x = 3, y = 2

x = \frac{c_1 b_2 - c_2 b_1}{a_1 b_2 - a_2 b_1} = \frac{12\cdot(-1) - 1\cdot3}{2\cdot(-1) - 1\cdot3} = 3,\quad y = \frac{a_1 c_2 - a_2 c_1}{a_1 b_2 - a_2 b_1} = 2

Try this: Each equation is a line. Where do the two lines cross? Check that point in both equations.
Try this: Change line 2 to 4x + 6y = 5. What happens to the crossing? Why is there no solution?
Try this: Now make line 2 equal to 4x + 6y = 24. How many solutions are there now?
Try this: Turn on Show the formula. The bottom number a₁b₂ − a₂b₁ is 0 exactly when the lines are parallel.

Name it

Concept 12.2.1 · Solving General Systems of Equations

Once you've found the system of equations in your word problem and/or the problem itself gives you a system of equation, you should consider one or more of the following methods for solving them.

Using Symmetry
Substituting Variables
Elimination
Adding/Subtracting/Multiplying Equations
Difference of Squares: $x^2 - y^2 = (x - y)(x + y)$
Binomial Expansions: $(x + y)^2 = x^2 + 2xy + y^2$ and $(x - y)^2 = x^2 - 2xy + y^2$
Other common factorization tricks (see algebraic number theory chapter for more)
Squaring equations if you have square roots
Introducing new variables and substituting
Graphing equation (Recommend graph paper)

Watch itpredict, then reveal

Example 12.1

If

\begin{aligned} 2x + y + z &= 36 \\ x + 2y + z &= 40 \\ x + y + 2z &= 32 \end{aligned}

Find $x + y + z$ .

1
Do you notice anything similar about the equations?

Predict first:

Example 12.2

EMCC

Suppose $x, y, z$ are real numbers that satisfy:

\begin{aligned} x + y - z &= 5 \\ y + z - x &= 7 \\ z + x - y &= 9 \end{aligned}

Find $x^2 + y^2 + z^2$ .

1
Add the first two equations. What happens to $x$ and $z$ ?

Predict first:
Your prediction

open on YouTube

Example 12.3

EMCC

Let $x$ , $y$ , and $z$ be real numbers so that $(x + y)(y + z) = 36$ and $(x + z)(x + y) = 4$ . Compute $y^2 - x^2$ .

1
What factor do the two equations share?

Predict first:

open on YouTube

Example 12.4

BMMT

Let $x$ and $y$ be real numbers such that $xy = 4$ and $x^2y + xy^2 = 25$ . Find the value of $x^3y + x^2y^2 + xy^3$

1
Factor $x^2y + xy^2$ . What is left after you pull out $xy$ ?

Predict first:
Your prediction

open on YouTube

Try it

Problem 12.4.2

AMC 8

In a jar of red, green, and blue marbles, all but 6 are red marbles, all but 8 are green, and all but 4 are blue. How many marbles are in the jar?

Problem 12.4.3

Omega Learn Math Contest

If

\begin{aligned} a \times b \times c &= 4 \\ b \times c \times d &= 6 \\ c \times d \times e &= 12 \\ d \times e \times f &= 18 \\ e \times a \times b &= 9 \end{aligned}

Find $a \times b \times c \times d \times e \times f$

Problem 12.4.5

I am thinking of 3 integers. Added two at a time their sums are 37, 41, 44. What is the product of the 3 integers?

Contest cornerreal contest problems

Problem 12.4.14

AMC 10

Let $a + 1 = b + 2 = c + 3 = d + 4 = a + b + c + d + 5$ . What is $a + b + c + d$ ?

Problem 12.4.15

AMC 10

If $y + 4 = (x - 2)^2$ , $x + 4 = (y - 2)^2$ , and $x \neq y$ , what is the value of $x^2 + y^2$ ?

Problem 12.4.16

EMCC

Given that

\begin{aligned} a + 5b + 9c &= 1 \\ 4a + 2b + 3c &= 2 \\ 7a + 8b + 6c &= 9 \end{aligned}

What is $741a + 825b + 639c$ ?

Problem 12.4.17(additional)

MATHCOUNTS

The units and tens digits of one two-digit integer are the tens and units digits of another two-digit integer, respectively. If the product of the two integers is 4930, what is their sum?

Problem 12.4.19(additional)

AMC 8

Four numbers are written in a row. The average of the first two is 21, the average of the middle two is 26, and the average of the last two is 30. What is the average of the first and last of the numbers?

Problem 12.4.24(additional)

MATHCOUNTS

The product of three distinct positive integers is 144. If the sum of the three integers is 26, what is the sum of their squares?

Checkpoint for section 12.2

Get 2 of 3 right to master this section. One try each, no hints.

12.30%

Word Problems

In this section you will be able to turn a word problem into equations by naming the unknowns, then solve.

A group buys 10 movie tickets for 24 dollars. Adults pay 3 dollars and kids pay 2 dollars. How many adults went? Two facts, two unknowns: that is a system of equations.

Word problems hide equations in sentences. Follow three steps:

Name each unknown with a letter.
Translate each fact into an equation.
Solve the system, then check your answer in the story.

Look:

“Three times as many” means multiply. “All but 6” means the rest add up to 6. Read slowly!

Exploreplay first, formulas later

Try this: A group buys a adult tickets at $3 and k kid tickets at $2. Line 1 says they bought 10 tickets. What does line 2 say?
Try this: Find a point on line 1 with whole numbers. Is it also on line 2?
Try this: Turn on Show the crossing. How many adult and kid tickets did they buy?

Name it

Concept 12.3.1 · System of Equations Word Problems

This is one of the most common topics on the AMC 8. The trick to solving these types of problems is to just assign variables to the unknowns in the problem and solve them.

Watch itpredict, then reveal

Example 12.5

Omega Learn

Orangey loves drinking diluted juice. Orangey takes an 8 ounce glass of orange juice and drinks some fraction of it. Then, he fills the rest with apple juice. After thoroughly mixing the glass, he drinks two thirds of the mixture. He then fills the rest with water. Orangey then finishes the whole glass. If he drank 4 times as much orange juice than apple juice, how much total liquid in ounces did he drink?

1
Orangey finishes the whole glass at the end. So how much orange juice does he drink in total?

Predict first:
Your prediction

open on YouTube

Example 12.6

AMC 8

Rectangles $R_1$ and $R_2$ , and squares $S_1$ , $S_2$ , and $S_3$ , shown below, combine to form a rectangle that is 3322 units wide and 2020 units high. What is the side length of $S_2$ in units?

1
Name the unknowns. Let the squares have sides $s_1, s_2, s_3$ . Look at the left edge and the middle column. How are $s_1$ and $s_2$ linked?

open on YouTube

Example 12.7

AIME

Jar A contains four liters of a solution that is $45\%$ acid. Jar B contains five liters of a solution that is $48\%$ acid. Jar C contains one liter of a solution that is $k\%$ acid. From jar C, $\frac{m}{n}$ liters of the solution is added to jar A, and the remainder of the solution in jar C is added to jar B. At the end both jar A and jar B contain solutions that are $50\%$ acid. Given that $m$ and $n$ are relatively prime positive integers, find $k + m + n$ .

1
Call the amount poured into jar A $x$ liters. Write the acid equation for jar A.

open on YouTube

Try it

Problem 12.4.6

AMC 8

Hui is an avid reader. She bought a copy of the best seller Math is Beautiful. On the first day, Hui read $\frac{1}{5}$ of the pages plus 12 more, and on the second day she read $\frac{1}{4}$ of the remaining pages plus 15 pages. On the third day she read $\frac{1}{3}$ of the remaining pages plus 18 pages. She then realized that there were only 62 pages left to read, which she read the next day. How many pages are in this book?

Problem 12.4.7

AMC 8

Before the district play, the Unicorns had won $45\%$ of their basketball games. During district play, they won six more games and lost two, to finish the season having won half their games. How many games did the Unicorns play in all?

Problem 12.4.12

AMC 10

Pablo, Sofia, and Mia got some candy eggs at a party. Pablo had three times as many eggs as Sofia, and Sofia had twice as many eggs as Mia. Pablo decides to give some of his eggs to Sofia and Mia so that all three will have the same number of eggs. What fraction of his eggs should Pablo give to Sofia?

Problem 12.4.22(additional)

AMC 10

At Megapolis Hospital one year, multiple-birth statistics were as follows: Sets of twins, triplets, and quadruplets accounted for 1000 of the babies born. There were four times as many sets of triplets as sets of quadruplets, and there was three times as many sets of twins as sets of triplets. How many of these 1000 babies were in sets of quadruplets?

Contest cornerreal contest problems

Problem 12.4.8

AMC 8

Ralph went to the store and bought 12 pairs of socks for a total of $24. Some of the socks he bought cost $1 a pair, some of the socks he bought cost $3 a pair, and some of the socks he bought cost $4 a pair. If he bought at least one pair of each type, how many pairs of $1 socks did Ralph buy?

Problem 12.4.11

AMC 10

Joe has a collection of 23 coins, consisting of 5-cent coins, 10-cent coins, and 25-cent coins. He has 3 more 10-cent coins than 5-cent coins, and the total value of his collection is 320 cents. How many more 25-cent coins does Joe have than 5-cent coins?

Problem 12.4.13

AMC 10/12

The sum of two natural numbers is 17,402. One of the two numbers is divisible by 10. If the units digit of that number is erased, the other number is obtained. What is the difference of these two numbers?

Problem 12.4.18(additional)

MATHCOUNTS

A car passes point A driving at a constant rate of 60 km per hour. A second car, traveling at a constant rate of 75 km per hour, passes the same point A a while later and then follows the first car. It catches the first car after traveling a distance of 75 km past point A. How many minutes after the first car passed point A did the second car pass point A?

Checkpoint for section 12.3

Get 2 of 3 right to master this section. One try each, no hints.

1

Tickets cost $3 for adults and $2 for kids. A group buys 12 tickets for $31. How many adult tickets did they buy?

2

At Megapolis Hospital one year, multiple-birth statistics were as follows: Sets of twins, triplets, and quadruplets accounted for 1000 of the babies born. There were four times as many sets of triplets as sets of quadruplets, and there was three times as many sets of twins as sets of triplets. How many of these 1000 babies were in sets of quadruplets?

3

Two jars each contain the same number of marbles, and every marble is either blue or green. In Jar 1 the ratio of blue to green marbles is 9 : 1, and the ratio of blue to green marbles in Jar 2 is 8 : 1. There are 95 green marbles in all. How many more blue marbles are in Jar 1 than in Jar 2?

12.4

Practice Problems

In this section you will be able to mix every equation trick on harder contest problems.

Ready for a challenge? These problems come from the AMC 8, 10 and 12. Name the unknowns, write the equations and look for a shortcut. Every problem has three hints if you get stuck.

Want every problem from this chapter on one page? Open the problem set.

Contest cornerreal contest problems

Problem 12.4.23(additional)

AMC 10

Two jars each contain the same number of marbles, and every marble is either blue or green. In Jar 1 the ratio of blue to green marbles is 9 : 1, and the ratio of blue to green marbles in Jar 2 is 8 : 1. There are 95 green marbles in all. How many more blue marbles are in Jar 1 than in Jar 2?

Problem 12.4.26(additional)

AMC 8

Each of the points A, B, C, D, E, and F in the figure below represents a different digit from 1 to 6. Each of the five lines shown passes through some of these points. The digits along each line are added to produce five sums, one for each line. The total of the five sums is 47. What is the digit represented by B?

Problem 12.4.27(additional)

AMC 12

The state income tax where Kristin lives is levied at the rate of $p\%$ of the first $28000 of annual income plus $(p + 2)\%$ of any amount above $28000. Kristin noticed that the state income tax she paid amounted to $(p + 0.25)\%$ of her annual income. What was her annual income?

Problem 12.4.28(additional)

AMC 10/12

Paula the painter and her two helpers each paint at constant, but different, rates. They always start at 8:00 AM, and all three always take the same amount of time to eat lunch. On Monday the three of them painted 50% of a house, quitting at 4:00 PM. On Tuesday, when Paula wasn't there, the two helpers painted only 24% of the house and quit at 2:12 PM. On Wednesday Paula worked by herself and finished the house by working until 7:12 P.M. How long, in minutes, was each day's lunch break?

Problem 12.4.29(additional)

AMC 12

A group of 12 pirates agree to divide a treasure chest of gold coins among themselves as follows. The $k^{\text{th}}$ pirate to take a share takes $\frac{k}{12}$ of the coins that remain in the chest. The number of coins initially in the chest is the smallest number for which this arrangement will allow each pirate to receive a positive whole number of coins. How many coins does the $12^{\text{th}}$ pirate receive?