Detailed Lesson on Equations
What is an equation?
An equation is a statement that two expressions are equal. The expressions can be made up of numbers, variables, and mathematical operations. The equal sign (=) is used to show that the expressions on either side of it are equal.
For example, the following are all equations:
- 2 + 3 = 5
- x + 5 = 10
- y^2 - 4 = 0
Solving equations
The goal of solving an equation is to find the value of the variable that makes the equation true. To do this, we need to manipulate the equation until the variable is isolated on one side.
There are many different ways to solve equations, but some of the most common methods include:
- Adding or subtracting the same number from both sides of the equation
- Multiplying or dividing both sides of the equation by the same number
- Using the distributive property
- Combining like terms
- Using factoring
Illustrations
Here are 6 illustrations of how to solve different types of equations:
Illustration 1: Solving a simple equation with one variable
Equation: 2x + 3 = 5
Solution:
- Subtract 3 from both sides of the equation:
2x + 3 - 3 = 5 - 3
- Simplify:
2x = 2
- Divide both sides of the equation by 2:
2x / 2 = 2 / 2
- Simplify:
x = 1
Therefore, the solution to the equation 2x + 3 = 5 is x = 1.
Illustration 2: Solving an equation with two variables
Equation: x + y = 10
Solution:
To solve an equation with two variables, we need to eliminate one of the variables. We can do this by adding or subtracting the equation from another equation.
In this case, let's subtract the equation x from both sides of the equation:
x + y - x = 10 - x
This simplifies to:
y = 10 - x
This means that y is equal to 10 minus the value of x. We can use this information to solve for y in any other equation that contains both x and y.
Illustration 3: Solving an equation with multiple terms
Equation: 2x + 3 = 5x + 2
Solution:
To solve an equation with multiple terms, we need to combine like terms on each side of the equation.
In this case, we can combine the x terms on the left side of the equation:
2x + 3 = **(5x + 2)**
(2 - 5)x + 3 = 2
-3x + 3 = 2
Next, we can subtract 3 from both sides of the equation:
-3x + 3 - 3 = 2 - 3
-3x = -1
Finally, we can divide both sides of the equation by -3:
-3x / -3 = -1 / -3
x = 1/3
Therefore, the solution to the equation 2x + 3 = 5x + 2 is x = 1/3.
Illustration 4: Solving an equation with a fraction
Equation: x + 1/2 = 3/4
Solution:
To solve an equation with a fraction, we can multiply both sides of the equation by the denominator of the fraction.
In this case, let's multiply both sides of the equation by 4:
(x + 1/2) * 4 = (3/4) * 4
This simplifies to:
4x + 2 = 3
Next, we can subtract 2 from both sides of the equation:
4x + 2 - 2 = 3 - 2
4x = 1
Finally, we can divide both sides of the equation by 4:
4x / 4 = 1 / 4
x = 1/4
Therefore, the solution to the equation x + 1/2 = 3/4 is x = 1/4.
Illustration 5: Solving an equation with a quadratic term
Equation: x
Illustration 5: Solving an equation with a quadratic term
Equation: x^2 + 2x - 3 = 0
Solution:
To solve an equation with a quadratic term, we can use the quadratic formula:
x = (-b ± √(b² - 4ac)) / 2a
where a, b, and c are the coefficients of the quadratic equation.
In this case, the coefficients are a = 1, b = 2, and c = -3.
Let's plug these values into the quadratic formula:
x = (-2 ± √(2² - 4 * 1 * -3)) / 2 * 1
x = (-2 ± √(16)) / 2
x = (-2 ± 4) / 2
x = 1 or x = -3
Therefore, the solutions to the equation x^2 + 2x - 3 = 0 are x = 1 and x = -3.
Illustration 6: Solving an equation with a radical
Equation: √x + 2 = 5
Solution:
To solve an equation with a radical, we need to isolate the radical on one side of the equation and then square both sides of the equation.
In this case, let's subtract 2 from both sides of the equation:
√x + 2 - 2 = 5 - 2
√x = 3
Next, we can square both sides of the equation:
(√x)² = 3²
x = 9
However, we need to check our answer to make sure that it is a valid solution to the original equation.
In this case, let's substitute x = 9 into the original equation:
√9 + 2 = 5
3 + 2 = 5
5 = 5
This equation is true, so we know that x = 9 is a valid solution.
Therefore, the solution to the equation √x + 2 = 5 is x = 9.
Video lessons
Here are some links to video lessons on solving equations:
- Solving linear equations: https://www.khanacademy.org/math/algebra-home/alg-basic-eq-ineq/alg-old-school-equations/v/algebra-linear-equations-1
- Solving quadratic equations: https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:quadratics-solve-factoring/v/example-1-solving-a-quadratic-equation-by-factoring
- Solving radical equations: https://www.khanacademy.org/math/algebra-home/alg-radical-eq-func
Conclusion
Equations are a fundamental part of mathematics. By learning how to solve equations, you will be able to solve many different types of problems.
- Algebra: Linear equations 1 | Linear equations | Algebra I | Khan Academy
- Algebra Basics: Solving Basic Equations Part 1 - Math Antics
- Algebra Basics: Solving 2-Step Equations - Math Antics
- Algebra - How To Solve Equations Quickly!
- Algebra Basics - Solving Basic Equations - Quick Review!
I hope this helps!
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