1. What is Linear?
2. Inverse Operations
3. One-step Linear Equations
4. Two-step Linear Equations
5. Multi-step Linear Equations
6. Practice Problems
Introduction
Can you imagine planning a road trip without understanding distances? Or baking your favorite dessert without grasping the concept of proportions? Both scenarios involve a form of linear equations. Yes, those daunting x’s and y’s actually play a crucial role in various aspects of our lives, from baking to road tripping, to more complex applications like engineering, economics, and computer science. This blog post aims to demystify the concept of linear equations, providing a step-by-step guide that transforms you into a confident solver of these important mathematical constructs.
Understanding Linear Equations; y = mx +b
A linear equation is an equation that creates a straight line when plotted on a graph. The most basic form of a linear equation is written as y = \color{red}m\color{black}x + \color{blue}b, where y is the dependent variable, x is the independent variable, m is the coefficient (or the slope of the line), and b is the constant (or the y-intercept).
The term mx, we call a linear term because the variable is raised to the first power; x=x^{1}. We call the b term a constant term because the value is constant and is not affected by any change in the input variable x. We say that this is a zero degree term; this is because we can think of b=b \cdot 1 = b \cdot x^{0}. As you can see, if we have just a number, that is equivalent to that number multiplied by x^{0}. Therefore, we will refer to the term with our input variable x as the linear term and the term without a variable as the constant term.
Example
\large{\begin{aligned}y &= \color{red}\frac{2}{3}\color{black}x+\color{blue}\pi \\ \color{red}m&=\frac{2}{3} \\ \color{blue}b&=\pi \end{aligned}}
Example
Remember that if there is no coefficient written, then the coefficient is equal to 1 and if there is no constant written, then the constant is equal to 0.
\large{\begin{aligned}y &= x \\ y &= \color{red}1\color{black}x + \color{blue}0 \\ \color{red}m&=1 \\ \color{blue}b&=0 \end{aligned}}
The primary goal when solving a linear equation is to isolate the variable of interest, which is often denoted as x. To achieve this, we employ the use of inverse operations. Inverse operations are mathematical procedures that undo or cancel out the effect of other operations, helping us to unravel the equation and isolate x.
Consider the following table of inverse operations:
\large{\begin{array}{|c|c|} \hline \text{Operation} & \text{Inverse Operation} \\ \hline \text{Addition (+)} & \text{Subtraction (-)} \\ \text{Subtraction (-)} & \text{Addition (+)} \\ \text{Multiplication (×)} & \text{Division (÷)} \\ \text{Division (÷)} & \text{Multiplication (×)} \\ \hline \end{array} }
These operations are considered inverse because they counteract the original operation, bringing the variable, typically an x, back to its “alone” or isolated state. In the following sections, we will delve deeper into one-step and two-step equations, and how to effectively utilize inverse operations to solve them.
One-Step Linear Equations
One-step equations are the simplest form of linear equations where you only need to perform one operation to solve for the variable. For example, in the equation “x+6=10“, you subtract 6 from both sides to isolate ‘x’, leading to “x = 4“. The most common mistake here is not performing the operation on both sides of the equation, which is essential to maintain the equality.
Example
Solve the equation for x \text{; }x + 5 = 11;
You would use the inverse operation of addition (which is subtraction) to remove the constant, 5, from the both sides of the equation.
\large{ \begin{aligned}x + 5 &= 11 \\ -5 &\text{ } -5 \\ x &= 6 \end{aligned}}
Example
Solve the equation for x \text{; }2x = 13;
You would use the inverse operation of multiplication (which is division) to remove the coefficient of the linear term, and this will result in x = 6.5.
\large{ \begin{aligned}2x &= 13 \\ \div2 &\text{ } \div2 \\ x &= 6.5 \end{aligned}}
Two-Step Linear Equations
Linear two-step equations involve two operations, a combination of addition or subtraction and multiplication or division. The main challenge here is determining the correct order of operations. Recall that when performing calculations in mathematics, we follow the GEMDAS rule (Grouping, Exponents, Multiplication/Division (left-to-right), Addition/Subtraction(left-to-right)). When we are solving equations we must “undo” the operations in the equation, so we must work in reverse order.
1. Locate the variable term (linear term)
2. Subtract or add the constant to isolate the variable term
3. Divide or Multiply by the coefficient of the linear term to isolate the variable
Example
Solve the equation for x \text{; }2x + 3 = 9;
\large{ \begin{aligned}2x +3&= 9 \\ -3 &\text{ } -3 \\ 2x &= 6 \\ \div2 &\text{ } \div2 \\ x &= 3 \end{aligned}}
Example
Now for a slightly more difficult example. Solve the equation for x \text{; }\frac{3}{5}x + \frac{2}{7} = \frac{7}{8};
\large{ \begin{aligned}\frac{3}{5}x + \frac{2}{7} &= \frac{7}{8} \\ -\frac{2}{7} &\text{ } -\frac{2}{7} \\ \frac{3}{5}x &= \frac{33}{56} \\ \div\frac{3}{5} &\text{ } \div\frac{3}{5} \\ x &= \frac{55}{56} \end{aligned}}
Multi-Step Linear Equations
As you advance, you’ll encounter equations that require multiple steps, combining like terms, applying the distributive property, or both. For instance, the equation 3(2x + 4) = 18 requires you to distribute the '3' to both terms in the parentheses first. Navigating these equations can be challenging, but with practice, they become more approachable.
Example;
Solve the equation for z \text{; } 4(z+3)=8
\large{ \begin{aligned}4(z+3) &= 8 \\ 4(z)+4(3) &= 8\\ 4z+12 &= 8 \\ -12 &\text{ } -12 \\ 4z &= -4 \\ \div4 &\text{ } \div4 \\ z &= -1 \end{aligned}}
Alternatively, you could start by dividing through by the 4.
\large{ \begin{aligned}4(z+3) &= 8 \\ \div4 &\text{ } \div4 \\ z+3 &= 2\\ -3 &\text{ } -3 \\ z &= -1 \end{aligned}}
Practice Makes Perfect
To truly master linear equations, practice is key. We’ve done a couple examples of one-step, two-step Resources like online equation solvers and practice websites can be great aids. A math tutor can also provide personalized guidance and help clarify any confusion. Try a range of exercises, and don’t shy away from seeking help if you find yourself struggling.
DOWNLOADS:
One-Step Equations
Version 1, Version 2, Version 3
Two-Step Equations
Version 1, Version 2, Version 3
Multi-Step Equations
Version 1, Version 2, Version 3
Conclusion
Mastering linear equations is one of the firsts steps in your journey. This skill will arm you with a valuable skill set that extends far beyond the classroom. As you venture further into the world of mathematics, you’ll find that the ability to understand and solve linear equations forms a bedrock of knowledge upon which many other concepts are built.
In the words of Pythagoras, “Numbers rule the universe.” Sodon’t just learn math; master it. The power to decipher the language of the universe is in your hands.
Keep practicing, stay curious, and above all, enjoy the journey.
