Algebra equations are equations that contain variables. Variables are symbols that represent unknown quantities. Algebra equations can be used to model many different real-world situations. For example, we can use algebra equations to calculate the price of items, the distance between two points, or the area of a triangle.
Types of Algebra Equations
There are many different types of algebra equations, but some of the most common include:
- Linear equations: Linear equations are equations of the form $ax + b = c$, where $a$, $b$, and $c$ are constants and $x$ is the variable.
- Quadratic equations: Quadratic equations are equations of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants and $x$ is the variable.
- Rational equations: Rational equations are equations that contain fractions.
- Radical equations: Radical equations are equations that contain radicals.
Solving Algebra Equations
There are many different ways to solve algebra equations, depending on the type of equation. Some of the most common methods include:
- Isolating the variable: This method involves using algebraic operations to manipulate the equation so that the variable is isolated on one side of the equation.
- Factoring: This method involves factoring the equation into two or more smaller equations, which are then easier to solve.
- Using the quadratic formula: This formula can be used to solve quadratic equations.
- Using graphing: This method involves graphing the equation and then finding the solutions where the graph crosses the $x$-axis.
Examples of Solving Algebra Equations
Here are some examples of how to solve different types of algebra equations:
Linear equation:
$2x + 3 = 7$
To solve this equation, we can isolate the variable $x$ by subtracting $3$ from both sides and then dividing both sides by $2$.
2x + 3 = 7
2x = 7 - 3
2x = 4
x = 2
Quadratic equation:
$x^2 + 2x – 3 = 0$
To solve this equation, we can factor it into two smaller equations: $(x + 3)(x – 1) = 0$. We then set each factor equal to zero and solve for $x$.
(x + 3)(x - 1) = 0
x + 3 = 0 or x - 1 = 0
x = -3 or x = 1
Rational equation:
$\dfrac{x}{x + 1} = \dfrac{1}{x – 2}$
To solve this equation, we can multiply both sides by the common denominator, $x(x + 1)(x – 2)$. This gives us the equation:
x = x - 2
0 = -2
This equation has no solution, so the rational equation has no solution.
Radical equation:
$\sqrt{x + 2} = 3$
To solve this equation, we can square both sides to get rid of the radical.
√x + 2 = 3
(√x + 2)^2 = 3^2
x + 2 = 9
x = 7
