How to Solve Math Problems: Non-Word Problems

Your teacher gives you homework, a test, or sends you to the board, to solve one or more problems. The types of problems we are talking about include, but not limited to, the following. These are somewhat mechanical problems (i.e., not word problems), but still require the student to understand the mathematical concepts to solve the problems.

Solve equations and inequalities
Simplify expressions
Factor polynomials
Graph equations and inequalities
Differentiate
Integrate
Matrices
Expand

Examples

Problem 1: Use elimination to solve the system for x. (Be careful! We want to solve for x NOT y.)

x – 4y = 1
x + 2y = 19

x = 1 + 4y
x = 19 – 2y, multiply by 2 to eliminate y.

x = 1 + 4y
2x = 38 – 4y

Add the two equations and solve for x.

3x = 39
x = 13 ∴ y = 3

Problem 2: Simplify the expression (x² – y²)/(x – y)

(x – y)(x + y)/(x – y), (x – y) cancels out in the numerator and denominator.

(x + y)

Problem 3: Factor the expression x² + 5x + 6

(x + 2)(x + 3)

Problem 4: Graph the inequalities x – y > 2, y > x²

Problem 5: Differentiate the expression x³ + 5x² -2x +1

3x² + 10x -2

Problem 6: Integrate the expression x³ + 5x² -2x +1

(x/12)(3x³ +20x² – 12x + 12) + C

Problem 7: Matrices dot product

Problem 8: Expand the expression (x + y)(x – y)

x² – y²

Additional Reading

“How to Graph an Equation”. 2022. wikiHow. https://www.wikihow.com/Graph-an-Equation.

Graphing equations is a much simpler process that most people realize. You don’t have to be a math genius or straight-A student to learn the basics of graphing without using a calculator. Learn a few of these methods for graphing linear, quadratic, inequality, and absolute value equations.

“Solving An Equation – Methods, Techniques, And Examples”. 2023. CUEMATH. https://www.cuemath.com/algebra/solving-an-equation/.

Solving equations involves finding the value of the unknown variables in the given equation. The condition that the two expressions are equal is satisfied by the value of the variable. Solving a linear equation in one variable results in a unique solution, solving a linear equation involving two variables gives two results. Solving a quadratic equation gives two roots. There are many methods and procedures followed in solving an equation. Let us discuss the techniques in solving an equation one by one, in detail.

The featured image on this page is from the YouTube video Algebra – How To Solve Equations Quickly!