2.2 Linear equations and systems

An expression tells us how to calculate a quantity. An equation says that two quantities are equal and asks when that statement is true. A system asks for values that make several equations true at the same time. These ideas appear whenever two descriptions of the same situation must agree: a total price, a measured distance, or two counts with a shared total.

This is foundational algebra. Review the expression rules in Module 2.1, then use OpenStax's one-variable equations before its two-variable systems. You need only substitution and arithmetic to start.

A basic idea: preserve the balance

Consider $2x+3=11$. The left and right sides have the same value only for certain $x$. Subtracting $3$ from both sides leaves $2x=8$; dividing both sides by $2$ gives $x=4$. Checking in the original equation gives $2(4)+3=11$.

An equation is like a balanced scale: adding or subtracting the same quantity on both sides preserves its solutions, as does multiplying or dividing by the same nonzero number. Dividing by a variable requires care because that variable might be zero. For example, dividing $x^2=x$ by $x$ would lose the valid solution $x=0$.

A quick tour of equations and systems

One unknown

A linear equation in one variable can be rearranged to $ax=b$. If $a\ne0$, it has the single solution $x=b/a$. Simplify brackets, gather the variable terms on one side, gather constants on the other, then check the result in the original equation.

If rearrangement instead produces $0=5$, there is no solution: no value can make a false statement true. If it produces $0=0$, every allowed value works. For instance, $2(x+1)=2x+2$ is true for all real $x$.

Two unknowns at once

In a system such as

$$x+y=5,\qquad 2x-y=1,$$

a solution is a pair $(x,y)$ that satisfies both equations. Adding the equations removes $y$: $3x=6$, so $x=2$ and then $y=3$. Substitution checks both: $2+3=5$ and $2(2)-3=1$.

Two common methods are:

Graphically, each two-variable linear equation is a line. The two graphs can cross once, run parallel and never meet, or coincide. Their system then has one, zero, or infinitely many solutions, respectively. The equations $x+y=5$ and $2x-y=1$ cross at $(2,3)$.

For more than two unknowns, the same idea continues: every equation must hold. Matrix methods in Module 11.4 organise larger systems.

Application 1: an equation with brackets

Solve $3(2x-1)=2x+13$.

1. Expand the left side. $6x-3=2x+13$.
2. Collect terms. Subtract $2x$ from both sides, then add $3$: $4x=16$.
3. Solve and check. $x=4$. The original sides are $3(8-1)=21$ and $2(4)+13=21$.

Checking in the original form catches a missed term when expanding.

Application 2: adult and child tickets

A group buys $17$ tickets. Adult tickets cost $8$ units, child tickets cost $5$ units, and the total paid is $109$ units. How many of each were bought?

1. Name the unknowns. Let $a$ be adult tickets and $c$ child tickets. Then $a+c=17$ and $8a+5c=109$.
2. Substitute. From the count equation, $c=17-a$. Thus $8a+5(17-a)=109$, or $3a=24$.
3. Find and check both counts. $a=8$ and $c=9$. They add to $17$, and $8(8)+5(9)=64+45=109$.

The count and cost equations tell us different things. Both are needed to determine the pair.