Solving for variables is a foundational concept in algebra and mathematics that helps us find the value of unknowns in equations. This lesson covers the basics of solving for variables, including linear equations, systems of equations, and real-life applications.
In algebra, a variable is a symbol (usually a letter) that represents an unknown value. An equation is a mathematical statement that asserts the equality of two expressions. Solving an equation for a variable means finding all values of the variable that make the equation true.
Single-step linear equations are the simplest form of equations where the variable can be isolated in one operation. The general form is \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants.
Example:
\(x + 5 = 12\)
To solve, subtract 5 from both sides of the equation:
\(x + 5 - 5 = 12 - 5\)
\(x = 7\)
Some equations require more than one step to isolate the variable. This involves using operations such as addition, subtraction, multiplication, and division.
Example:
\(2x - 3 = 11\)
First, add 3 to both sides to get rid of the -3:
\(2x = 14\)
Then, divide by 2 to isolate \(x\):
\(x = 7\)
Equations may have variables on both sides. The goal is to get all the variables on one side and the constants on the other.
Example:
\(3x + 4 = 2x + 10\)
Subtract \(2x\) from both sides:
\(x + 4 = 10\)
Subtract 4 from both sides to isolate \(x\):
\(x = 6\)
When equations include fractions, the approach to solving them remains the same, but it may involve additional steps like finding a common denominator or multiplying both sides of the equation by the least common multiple to eliminate fractions.
Example:
\(\frac{1}{2}x + 3 = 7\)
Multiply everything by 2 to eliminate the fraction:
\(x + 6 = 14\)
Subtract 6 from both sides:
\(x = 8\)
When there are multiple equations with multiple variables, we have a system of linear equations. The goal is to find the values of the variables that satisfy all equations in the system.
There are several methods for solving systems of equations, including substitution, elimination, and graphing. We will look at the substitution and elimination methods.
The substitution method involves solving one of the equations for one variable and then substituting that expression into the other equation.
Example:
\(System: \begin{cases} x + y = 6\ x - y = 2 \end{cases}\)
Solve the first equation for \(x\):
\(x = 6 - y\)
Substitute \(x\) in the second equation:
\(6 - y - y = 2\)
Solve for \(y\):
\(2y = 4\)
\(y = 2\)
Substitute \(y\) back into \(x = 6 - y\):
\(x = 4\)
The elimination method involves adding or subtracting the equations to eliminate one of the variables.
Example:
\(System: \begin{cases} x + y = 6\ x - y = 2 \end{cases}\)
Add the equations to eliminate \(y\):
\(2x = 8\)
Solve for \(x\):
\(x = 4\)
Substitute \(x\) back into one of the original equations to solve for \(y\):
\(4 + y = 6\)
\(y = 2\)
Solving for variables is not just an academic exercise but has practical applications in everyday life, from calculating distances, speed, and time in travel, to budgeting finances, and even in more complex fields like engineering and physics.
Understanding how to manipulate and solve equations allows us to make predictions and understand relationships between different quantities in our world.
