- Linear equations are one of the most important mathematical models you use to describe relationships where change is constant.
- In this topic, you will focus on solving linear equations in a way that is logically justified, using equivalence transformations (steps that keep the solution set the same), and you will connect algebraic solutions to graphical ones.
Linear Equations Describe A Balance
A linear equation is an equation in which the variable(s) appear only to the power 1, and are not multiplied together.
- $3x-5=2$
- $6-2x=11$
- $\frac14 x=3$
- $7x-1=2x+8$
What makes an equation an equation (not just an expression) is the equals sign, which represents a balance: both sides have the same value.
Solution (of an equation)
A value of the variable that makes the equation true when substituted into the equation.
Equivalent Equations
Equations that have exactly the same solutions.
The key idea in solving linear equations is to perform operations that produce equivalent equations, so you never "change the answer set" while you simplify.
Equivalence Transformations Preserve Solutions
- An equivalence transformation is a change you make to an equation that results in an equivalent equation (same solution set).
- The most common equivalence transformations are based on two principles.
Addition principle
If you add (or subtract) the same quantity to both sides of an equation, the resulting equation is equivalent to the original.
Multiplication principle
If you multiply (or divide) both sides of an equation by the same non-zero quantity, the resulting equation is equivalent to the original.
These principles justify the standard steps you already use, such as "add 5 to both sides" or "divide both sides by 3".
- Dividing (or multiplying) both sides by 0 is not an equivalence transformation.
- This is simply because multiplying both sides by 0 destroys information.
- For example, from $x=2$ you can multiply both sides by 0 to get $0=0$, but $0=0$ is true for every value of $x$, so you changed the solution set.
Distributive And Simplifying Steps Still Matter
- Not every step you do is an "addition" or "multiplication" principle step.
- Sometimes you are using algebraic properties to rewrite expressions without changing their value.
- For example, expanding brackets uses the distributive property: $$2(3x-4)=6x-8$$
- That rewriting does not change the equation, it just expresses the same left-hand side more simply.
When you write solutions, make each step an equation (with an equals sign) so it is clear that you are producing equivalent equations at every stage.
A Reliable Algebraic Method For Solving Linear Equations
Most linear equations can be solved by a consistent plan:
- Simplify each side (expand brackets, clear fractions if helpful).
- Use the addition principle to gather all variable terms on one side and constants on the other.
- Use the multiplication principle to make the coefficient of the variable equal to 1.
- Check by substituting your solution back into the original equation.
Solve $2(3x-4)=3x+7$.
Solution
- Step 1: Expand the left side (distributive property): $$6x-8=3x+7$$
- Step 2: Use the addition principle to remove $3x$ from the right by subtracting $3x$ from both sides: $$3x-8=7$$
- Step 3: Use the addition principle to remove $-8$ by adding 8 to both sides: $$3x=15$$
- Step 4: Use the multiplication principle (divide both sides by 3): $$x=5$$
- Check:
- Left side: $2(3\cdot 5-4)=2(15-4)=22$
- Right side: $3\cdot 5+7=15+7=22$
- So $x=5$ is correct.
A fast, clear check is often worth doing even if not explicitly asked, because it catches sign errors (especially with negatives and brackets).
Fractions: Keep Them Or Clear Them Carefully
- Equations with fractions are still solved with the same principles, but you must stay organized.
- Two common approaches are:
- Work with fractions directly (often fine if denominators are simple).
- Clear denominators by multiplying every term by the lowest common multiple (LCM) of denominators.
- If you clear fractions, you must multiply the entire equation (both sides) by the same non-zero number.
- Multiplying only one term breaks equivalence.
Solve $\frac{x}{3}+2=\frac{1}{2}x-4$.
Solution
- Lowest common multiple of 3 and 2 is 6. Multiply both sides by 6: $$6\left(\frac{x}{3}+2\right)=6\left(\frac{1}{2}x-4\right)$$
- Simplify: $$2x+12=3x-24$$
- Now use addition principle: $$12=x-24 \Rightarrow x=36$$
- Check (quick):
- LHS: $\frac{36}{3}+2=12+2=14$
- RHS: $\tfrac12\cdot 36-4=18-4=14$
Some Operations Do Not Create Equivalent Equations
- Equivalence transformations are special because they preserve solutions.
- Some operations can change the solution set.
- Examples of operations that are not always equivalence transformations:
- Squaring both sides (can introduce extra solutions). Example: from $x=-2$ you get $x^2=4$, but $x^2=4$ has two solutions $x=\pm 2$.
- Taking square roots without considering both signs.
- Multiplying both sides by an expression involving the variable that could be zero, without tracking the case when it is zero.
- In linear equations, you usually only multiply or divide by non-zero constants, so equivalence is straightforward.
- Later, with quadratic and rational equations, you must be much more careful.
Graphical Meaning: Solving An Equation As An Intersection
- A powerful modelling idea is that any equation $$\text{(left side)}=\text{(right side)}$$ can be turned into a system by setting each side equal to $y$: $$y=\text{(left side)}\qquad\text{and}\qquad y=\text{(right side)}$$
- The solution of the original equation is the x-coordinate where the two graphs intersect, because that is where the two sides are equal.
Solve $3(x+2)-6=4(2x-3)+1$.
Solution
Graph:
- $y=3(x+2)-6$
- $y=4(2x-3)+1$
The intersection gives the solution.
From the intersection point on the graph, the solution is: $$x=\frac{11}{5}$$
(And the common $y$-value there is $\frac{3}{5}$.)
- Graphing is especially useful to check whether your algebraic answer is reasonable.
- If your algebra says $x=10$ but your intersection is near $x=2$, you know something went wrong.
Linear Equations As Models In Real Contexts
In the IB MYP, linear equations are often used as models in scientific and technical contexts.
A typical modelling workflow is:
- Define the variable(s) clearly.
- Build an equation from the context.
- Solve using equivalence transformations.
- Interpret the solution in context.
- Decide whether the mathematical solution is realistic (does it make sense with units and constraints?).
- A technician compares two pricing plans for a device subscription.
- Plan A: fixed fee plus cost per month: $C=15+8m$
- Plan B: different fixed fee and cost per month: $C=5+10m$
- To find when they cost the same, solve: $$15+8m=5+10m$$
- This is a linear equation, and the solution tells you the break-even month.
- The mathematics is exact, but you still must interpret whether $m$ should be a whole number of months and whether negative values make sense.
- Name the two main principles used to create equivalent equations.
- Explain why "multiply both sides by 0" does not preserve solutions.
- For the equation $7x-1=2x+8$, which principle do you use first to isolate $x$?
- If two lines do not intersect, what does that mean about the equation formed by setting their $y$-values equal?
