define/illustrate expression, equation and order of operations
Expression: In mathematics, an expression is a combination of numbers, variables, and operators that are written in a specific sequence. It does not contain an equal sign and may or may not have a specific value. For example, 3x + 5y is an expression, as it consists of variables (x and y), coefficients (3 and 5), and an operator (+).
Equation: An equation is a mathematical statement that shows the equality of two expressions. It contains an equal sign and implies that both sides of the equation have the same value. Equations are often used to solve for unknown variables and find their values. For example, the equation 2x + 7 = 15 represents that the expression 2x + 7 is equal to 15. The goal is to determine the value of x that satisfies the equation.
Order of Operations: The order of operations is a set of rules that determines the sequence in which various operations are performed in a mathematical expression. The acronym PEMDAS is commonly used to remember the order of operations. It stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). These rules help ensure that calculations are done accurately and consistently.
Illustration:
Consider the expression: 3 + 4 × 2
Using the order of operations (PEMDAS):
1. We first perform the multiplication: 4 × 2 = 8
2. Then, we perform the addition: 3 + 8 = 11
Therefore, the value of the expression 3 + 4 × 2 is 11.
Now, let's take an equation: 2x + 3 = 11
To solve for x, we isolate the variable:
1. Subtract 3 from both sides: 2x = 11 - 3 = 8
2. Divide both sides by 2: x = 8 ÷ 2 = 4
Thus, the solution to the equation 2x + 3 = 11 is x = 4.
GEOLOGY ROCKS: A geologist and has a laboratory full of rocks. He knows that each rock weighs exactly one pound (+1), and he would like to figure out how many rocks are in each crate. He places crates and rocks on a scale until they are balanced. Using his math skills, he is able to reason how many rocks are in each crate without having to look inside. Write an equation. Can you use mental math to see how many rocks are in each crate for each scenario?
Let's assume there are two crates, with x and y rocks in each crate respectively.
Since each rock weighs one pound, the total weight of the rocks in the first crate would be x pounds, and the total weight of the rocks in the second crate would be y pounds.
As the geologist wants to figure out how many rocks are in each crate, he balances the crates on a scale. The equation representing the balance would be:
x = y
By setting the weight of the rocks in each crate equal, the geologist can determine that the number of rocks in each crate is the same.
To mentally determine how many rocks are in each crate, we can consider different scenarios:
Scenario 1: If the geologist places a crate with 5 rocks on one side of the scale, and a crate with 5 rocks on the other side, the weight will balance. Therefore, x = y = 5, meaning there are 5 rocks in each crate.
Scenario 2: If the geologist places a crate with 10 rocks on one side, and a crate with 10 rocks on the other side, the weight will balance. Hence, x = y = 10, indicating there are 10 rocks in each crate.
By observing the balance of the scale, the geologist can mentally determine the number of rocks in each crate based on the equation x = y, without having to look inside the crates.
Think of 4x - 1 = 15. What value of x would make 4x - 1 equal to 15?
To find the value of x that makes the equation 4x - 1 equal to 15, we can solve the equation by isolating the variable.
Given: 4x - 1 = 15
First, we want to get rid of the -1 on the left side of the equation. We can do this by adding 1 to both sides:
4x - 1 + 1 = 15 + 1
This simplifies to:
4x = 16
Next, we want to isolate x, so we divide both sides of the equation by 4:
4x/4 = 16/4
This simplifies to:
x = 4
Therefore, the value of x that would make 4x - 1 equal to 15 is x = 4.
Expression:
An expression is a combination of numbers, variables, and operators that are grouped together but does not include an equal sign. It can be as simple as a single number or variable, or it can be more complex with multiple variables and operators. For example, 3x + 5 is an expression.
Equation:
An equation is a mathematical statement that uses an equal sign (=) to show that two expressions are equal. It consists of two sides, the left-hand side and the right-hand side, separated by the equal sign. The left-hand side and right-hand side of an equation can contain expressions with variables, numbers, and operators. For example, 2x + 3 = 7 is an equation.
Order of Operations:
The order of operations is a set of rules that determines the sequence in which mathematical operations should be performed in an expression or equation. The standard order of operations is as follows:
1. Parentheses: Perform operations inside parentheses first.
2. Exponents: Next, evaluate any exponentiation or powers.
3. Multiplication and Division: Perform multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, perform addition and subtraction operations from left to right.
These rules are used to clarify the order in which calculations should be carried out in an expression or equation to ensure consistent and accurate results.
Expression: An expression is a combination of numbers, variables, and mathematical operations, without an equal sign. It represents a mathematical calculation or a value. For example, "2 + 3" or "4x - 6" are expressions.
Equation: An equation is a mathematical statement that contains an equal sign and shows that two expressions or values are equal. It consists of two expressions on either side of the equal sign. For example, "2x + 5 = 13" or "3y^2 - 7 = 2y + 11" are equations.
Order of Operations: The order of operations, also known as PEMDAS or BODMAS, is a set of rules that determine the order in which mathematical operations should be performed in an expression or equation. It stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This order ensures that the calculations are done correctly and results in a unique value. This is necessary because without the correct order, different individuals may come up with different answers for the same mathematical expression.
To illustrate how to use the order of operations, let's consider the expression: 2 + 3 × 4 / ( 2 - 1 ).
First, we identify the parentheses: ( 2 - 1 ) = 1.
Next, we evaluate the multiplication and division from left to right: 3 × 4 = 12, and 12 / 1 = 12.
Finally, we perform the addition and subtraction from left to right: 2 + 12 = 14.
Therefore, the answer to the expression is 14.