By 7th grade, math students are well and truly familiar with algebra. They’ve been introduced to equations, variables, and other important concepts in algebra. Before they can start thinking about solving advanced math problems, like SAT-level linear equations, or high school quadratic equations, students need to get comfortable with the basics.
Linear equations with one variable are a fantastic starting point for students that will eventually be able to solve these more difficult problems mentioned above. Before math students can solve linear equations with one variable, they must adequately understand these terms, and learn the method that must be followed to find the answer.
Linear equations
Let’s start at the beginning. Linear equations are key mathematical tools used to represent the relationship between variables. When we graph linear equations on the coordinate plane, (the x-y axes) the result is a straight line – hence their name “linear”, which is defined as something relating to a straight line; or having only one dimension.
Linear equations are of fundamental importance to professionals in many fields beyond mathematics, including physicists, physicians, architects, and more. Similarly, linear equations can be solved using a couple of different methods.
Linear equations can contain any multiple number variables, but for the purposes of this article, we are going to focus specifically on linear equations with just one variable.
Linear equations with one variable
Another way to define a linear equation with just one variable is to call it an equation that can be written in the form “ax + b = 0”, where ‘a’ and ‘b’ are constants, and ‘x’ is a variable. When we are faced with these equations, our goal is to identify the value of the variable ‘x’ that makes the statement true. For example, if our equation was: 3x + 0 = 3, we could say that 3x = 3, therefore x must equal 1.
As mentioned, linear equations play a crucial role in many different fields. From economics to engineers to our everyday DIY tasks, there are few jobs entirely unaffected by linear equations. Physicists use these equations to talk about the behavior of physical systems, motion, electrical circuits, and more. Economists use them to describe supply and demand, economic growth, and market characteristics. Engineers need to be able to solve linear equations to better analyze forces, design structures, and optimize processes.
In essence, by solving linear equations, we are able to find the intersection point of lines, make predictions based on mathematical models, and analyze data trends. On top of this, linear equations touch us regularly in our everyday lives. They are used to solve real-world problems, as well as planning, budgeting, and other vital aspects of our lives.
Solving linear equations with one variable
To solve linear equations with one variable, we must follow a careful process of algebraic operations. As we are trying to solve for our one variable ‘x’, we want to isolate this variable on one side of the equation.
Let’s take a closer look with a straightforward example:
3x + 3 = 12
First, we isolate the variable. We do this by subtracting 3 from each side of the equals sign, leaving us with:
3x – 3 = 12 – 3
3x = 9
Next, we need to turn 3x into x, so we divide both sides of the equals sign by 3:
3x/3 = 9/3
x = 3
Now we can see that our answer is x = 3.
In some cases, linear equations may have more than one unique solution or infinite solutions. Consider the problem 3x – 6 = 3x + 2. This particular equation does not yield a valid solution for ‘x’, because when we simplify the equation we get -6 = 2. In these instances, we say that the equation is inconsistent, basically meaning that there is no value of ‘x’ that satisfies the original equation.
If we want to get faster at solving linear equations with one variable, we must learn the following process:
- Isolate the term containing the variable on one side of the equation;
- Use algebraic operations to simplify both sides of the equation;
- Continue simplifying until the variable is isolated;
- Identify if the equation has a unique solution, no solution, or infinitely many solutions;
- To check: linear equations should always be a straight line when graphed.
At Online Math Center
At Online Math Center, we understand how many variables there are in a child’s math education. A fundamental understanding of mathematical concepts is a key skill that all students should aspire to have, but we understand that this isn’t always possible in school classrooms filled to the brim with students.
We offer individual tutoring classes to help plug the gaps in a child’s knowledge and get ahead of their classmates with lessons focused on that child’s individual strengths, weaknesses, and educational needs.
Contact OMC today to give your child a head start in math class.
