Using equations and variables can help us modeling different situations. They can also help us solving problems where there are unknown values. Let's consider the cases where there are one and two unknowns.

One-Variable Equations

We can solve a problem where there is an unknown value using a one-variable equation. For this, we need to write the verbal information given in the problem as a mathematical expression and set an equation. Solving this equation allows us to find the unknown value.

Peter is 10 years older than Jamie . Their combined ages add up to 40 .

Let's use a table.

Verbal Information	Algebraic Expression / Equation
Jamie's age	$x$
Peter is $10$ years older than Jamie	$x + 10$
Peter and Jamie's ages add up to $40 .$	$x + (x + 10) = 40$

Then, our equation to solve would be $x + (x + 10) = 40 .$ Notice that the equation is true only when $x = 15 .$

Extra

Solving the Equation

Let's solve the equation.

x + (x + 10) = 40

▼

Solve for

x

RemovePar

Remove parentheses

x + x + 10 = 40

AddTerms

Add terms

2 x + 10 = 40

SubEqn

$LHS - 10 = RHS - 10$

2 x = 30

DivEqn

$LHS / 2 = RHS / 2$

x = 15

The solution of the equation is

15 .

Two-Variable Equations

A two-variable equation can help us model a situation where a quantity depends on another following a specific rule or pattern. We can represent such relationship as a mathematical expression.

Miles covered	Gallons of Gas Used
$40$	$1$
$80$	$2$
$120$	$3$

There is a clear pattern between the number of miles covered and the gallons of gas used. For each gallon used,

40

miles are covered. Then, if we use

x

to represent the number of gallons of gas used, and

y

for the number of miles covered the relationship can be represented by an equation.

y = 40 x

From the above table, we know that the ordered pairs

(1, 40),

(2, 80),

and

(3, 120)

make the equation true. Therefore, they are some of the solutions of

y = 40 x .

Another Type of Two-Variable Equations

We can also solve problems where there are two unknown values using two-variable equations by writing and solving a System of Linear Equations.

{Equation Equation (I) (II)

Solving these equations as a system allows us to find the unknown values. Let's consider an example.

In a group of 40 students, the number of girls is ten more than twice the number of boys .

Let's use a table.

Verbal Information - First Equation	Algebraic Expression - First Equation
Number of boys	$x$
Number of girls	$y$
There are $40$ students in total	$x + y = 40$
Verbal Information - Second Equation	Algebraic Expression - Second Equation
Twice the number of boys	$2 x$
The number of girls is $10$ more than twice the number of boys.	$y = 2 x + 10$

We can write the system formed by the above equations.

{x + y = 40 y = 2 x + 10 (I) (II)

Notice that for this kind of problem, we need two equations in order to find the unknown values. This system has only one solution which is the ordered pair

(10, 30) .

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Solving the System

To solve the system, we can substitute

2 x + 10

for

y

in Equation (I). This will reduce the number of variables in Equation (I) to one.

{x + y = 40 y = 2 x + 10

Substitute

$(I):$ $y = 2 x + 10$

{x + 2 x + 10 = 40 y = 2 x + 10

▼

(I):

Solve for

x

AddTerms

$(I):$ Add terms

{3 x + 10 = 40 y = 2 x + 10

SubEqn

$(I):$ $LHS - 10 = RHS - 10$

{3 x = 30 y = 2 x + 10

DivEqn

$(I):$ $LHS / 3 = RHS / 3$

{x = 10 y = 2 x + 10

Next by substituting

x = 10

in Equation II, we can find

y -

value.

{x = 10 y = 2 x + 10

Substitute

$(II):$ $x = 10$

{x = 10 y = 2 (10) + 10

▼

(II):

Solve for

y

Multiply

$(II):$ Multiply

{x = 10 y = 20 + 10

AddTerms

$(II):$ Add terms

{x = 10 y = 30

Therefore, the ordered pair

(10, 30)

makes the system true. Note that is the only solution.

Conclusion

Now, let's compare and contrast the two type of equations.

Similarities

One- and two-variable equations can help us solving problems where there are unknown value(s).
Both type of equations may have solution(s).

Differences

Since one-variable equations contain one variable, the value of the variable can be found using one equation.
A two-variable equation models the relationship between two quantities.
A solution of a two-variable equation is of the form $(x, y) .$
Two-variable equations need two equations to be solved as a system.

Exercise

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