Equations

Many pipefitting problems involve equations or formulas, making them important tools for solving trade-related tasks. It is essential to understand how they work.

An equation is a mathematical sentence that uses an equal sign (=) to show that two expressions are equal, for example [latex]2x = 12[/latex]. A formula is a type of equation that shows a relationship between quantities. Ohm's Law, for example, describes the relationship between voltage, amperage, and resistance in a circuit:

[latex]\quad\text{Voltage} = \text{Amperage} \times \text{Resistance}[/latex]

Equations can include variables (unknown values) and constants (known values). Solving an equation means finding the value of the variable that makes the equation true.

You can think of an equation like a balance scale—whatever you do to one side, you must do to the other to keep it balanced. The process of isolating the variable on one side of the equal sign is called transposing.

Key idea: Whatever you do to one side of the equation, you must do to the other side.

We will first use this “balance method” to solve equations, and later learn a shortcut called transposing.

Order of Operations (BEDMAS)

Before solving equations or working with formulas, it is important to understand the correct order for completing calculations. This is called the order of operations, and it ensures that everyone solves problems in the same way and gets the same answer.

Many tradespeople use the acronym BEDMAS to help them remember the correct order:

STEP 1 → Brackets ( )
STEP 2 → Exponents 2³
STEP 3 → Division ÷
STEP 4 → Multiplication ×
STEP 5 → Addition +
STEP 6 → Subtraction −

How BEDMAS Works

When solving a problem, follow these steps:

1. Brackets first – solve anything inside brackets ( ), [ ], or { }.
2. Exponents next – solve powers (such as squared numbers).
3. Division and multiplication – work from left to right.
4. Addition and subtraction – work from left to right.

Important: Division and multiplication are done in order from left to right. The same rule applies to addition and subtraction.

Example 1:

[latex]\quad 6 + 2 \times 3[/latex]

Step 1: Multiply first (because of BEDMAS)
[latex]\quad 2 \times 3 = 6[/latex]

Step 2: Add
[latex]\quad 6 + 6 = 12[/latex]

Answer: 12

Example 2 (using the same numbers as above):

[latex]\quad (6 + 2) \times 3[/latex]

Step 1: Solve inside the brackets
[latex]\quad 6 + 2 = 8[/latex]

Step 2: Multiply
[latex]\quad 8 \times 3 = 24[/latex]

Answer: 24

The brackets change the answer!

Example 3:

[latex]\quad 20 \div 5 + 3[/latex]

Step 1: Divide (left to right)
[latex]\quad 20 \div 5 = 4[/latex]

Step 2: Add
[latex]\quad 4 + 3 = 7[/latex]

Answer: 7

Why BEDMAS Matters

In the trades, calculations are often part of formulas and measurements. If the order of operations is not followed correctly, the final answer may be wrong, which can lead to errors in measurements, materials, or system performance.

By following BEDMAS, you can ensure your calculations are accurate and consistent.

Grouping Numbers

Parentheses ( ) and brackets [ ] mean that two or more numbers are to be treated as one. For practical purposes, they indicate that the work in parentheses is to be done first. [latex]2(a + b)[/latex]  reads “two times the sum of a plus b,” and [latex]2a + b[/latex]  reads “two times a plus b.” 

If [latex]a = 5[/latex] and [latex]b = 6[/latex], then [latex]2(a + b) = 22[/latex] and [latex]2a + b = 10 + 6 = 16[/latex].

A long division bar groups numbers: [latex]\frac{a+b}{5}[/latex] means that the sum of a  and b are divided by 5.

However, [latex]\frac{a}{5}+b[/latex] means that only a is divided by 5.

Again, if a = 5 and b = 6, then [latex]\frac{5+6}{5}=\frac{11}{5}=2\frac{1}{5}[/latex].

But, [latex]\frac{5}{5}+6=1+6=7[/latex].

A long radical bar such as [latex]\sqrt{a^2+b^2}[/latex] indicates the square root of the sum of the two squares.

Change subtraction into addition:

Example:

4 − 6 = 4 + (−6)

Multiplication Rules

The sign of the product:

• Same signs → positive
• Different signs → negative

Examples:

[latex]\quad\begin{array}{ccc} &+&3\ \times&+&3\\hline &+&9 \end{array}[/latex]

[latex]\quad\begin{array}{ccc} &+&4\ \times&-&3\\hline &-&12 \end{array}[/latex]

[latex]\quad\begin{array}{ccc} &-&4\ \times&-&4\\hline &+&16 \end{array}[/latex]

Division Rules

• Same signs → positive
• Different signs → negative

[latex]\quad (+4) \div (+2) = +2 \qquad (+12) \div (-3) = -4[/latex]
[latex]\quad (-6) \div (-3) = +2 \qquad (-16) \div (+4) = -4[/latex]

Solving Simple Equations (Balance Method)

Example of an Addition Equation

[latex]\quad x + 15 = 30[/latex]

To solve the equation, we must try to get the variable x alone on one side. Use the inverse of adding 15—or subtracting 15—to get x alone on the left side. Now x is alone on the left side, since 15 − 15 = 0, but the scale is not in balance. To balance the scale, subtract 15 from the right side of the equation.

[latex]\quad x + 15 - 15 = 30 - 15[/latex]
[latex]\quad x = 15[/latex]

Example of a Subtraction Equation

[latex]\quad y - 9 = 3[/latex]

To solve this equation, we must try to get the variable y alone on one side. We can use the inverse of subtracting 9, or adding 9, to get y alone on the left side.

[latex]\quad y - 9 + 9 = 3 + 9[/latex]

[latex]\quad y - 0 = 12[/latex]

[latex]\quad y = 12[/latex]

Example of a Multiplication Equation

The mechanical contract to construct the regional hospital was estimated at $4,775,000.00, which was 4 times the cost of the administration building in the same complex. What was the cost of the administration building contract?

We can write a multiplication equation to find the answer to problems like this. Our unknown number is the cost of the administration building contract. Let x represent this cost. We know that 4x is the cost of the regional hospital. We can write our equation like this:

[latex]\quad4x = $4,775,000[/latex]

To solve this equation, we can use the inverse of multiplying by 4, which is dividing by 4.

If we divide the left side of the equation by 4, we will get x alone on the left. Remember, any operation done to one side must also be done to the other side, so we must also divide the right side by 4:

Checking Equations

Always check your answer by substituting it back into the original equation.

The process of checking is essential at all times to prove the accuracy of dimensions and quantities. It is comparatively simple to substitute computed values into the original equation to determine the accuracy of computation.

Substitute the computed value of any letter or symbol into the original equation. Perform each operation as indicated.

Note: The equation is balanced when the values on both sides of the equation are equal.

Example 1:

Check the answer [latex]b = 9[/latex] in the equation [latex]6b = 54[/latex].

First substitute the value of [latex]b[/latex] in the original equation, [latex]6b = 54[/latex].

[latex]\quad 6 \times 9 = 54[/latex]

Since both sides are equal, the answer is correct.

Example 2:

Solve the equation [latex]4x + 3 = -2x + 33[/latex].

First, transpose the known quantity 3 from left to right in the original equation and change the sign:

[latex]\quad4x = -2x + 33 - 3[/latex]

Next, transpose the unknown quantity from right to left and change the sign:

[latex]\quad4x + 2x = 33 - 3[/latex]

Next, collect all terms:

[latex]\quad6x = 30[/latex]

Finally, divide both sides by 6:

[latex]\quad x = \frac{30}{6}[/latex]

Answer:

[latex]\quad x = 5[/latex]

Check: Substitute the computed value 5 for the unknown x in the original equation:

[latex]\quad4x + 3 = -2x + 33[/latex] (Original equation)

[latex]\quad4(5) + 3 = -2(5) + 33[/latex]

Next, collect terms on both sides of the equation by performing the operations as indicated. Finally, check final values on both sides of the equation to see that they are equal:

[latex]\quad23 = 23[/latex]

Transposing with Addition

[latex]\quad x + 15 = 92[/latex]
[latex]\quad x = 92 - 15[/latex]
[latex]\quad x = 77[/latex]

Transposing with Subtraction

[latex]\quad y - 45 = 81[/latex]

If we move the –45 across the equal sign, it becomes +45 on the other side.
As before, we must eliminate the –45, which would isolate y by itself on one side of the equal sign:
[latex]\quad y - 45 = 81[/latex]

[latex]\quad y = 81 + 45[/latex]

[latex]\quad y = 126[/latex]

Transposing with Multiplication

The area of a square or rectangle is expressed in square units and is the product of the length times the width. The equation is expressed as follows:

[latex]\quad\text{Area} = \text{Length} \times \text{Width}[/latex]

If you are asked to solve for the width of a rectangle, the equation can be transposed as follows: Isolate the width by moving the length across the equal sign. Because it was multiplication on the right, it becomes division on the left.

[latex]\quad\frac{\text{Area}}{\text{Length}} = \text{Width}[/latex]

If you are asked to solve for the length of a rectangle, isolate the length by moving the length across the equal sign and divide.

[latex]\quad\frac{\text{Area}}{\text{Width}} = \text{Length}[/latex]

Transposing with Division

The pressure found in piping systems is expressed as the force (pounds) pushing on a unit of area (usually a square inch). We shortcut this relationship by saying psi. The actual equation is expressed as follows:

[latex]\quad\text{Pressure} = \frac{\text{Force}}{\text{Area}}[/latex]

If you are asked to solve for the force in pounds or the area in square inches, you must isolate the variable you are looking for by moving the known value across the equal sign. If the known variable was the numerator, it becomes the denominator, and vice versa.

[latex]\quad\text{Pressure} = \frac{\text{Force}}{\text{Area}} \rightarrow \frac{\text{Force}}{\text{Pressure}} = \text{Area}[/latex]

[latex]\quad\quad \quad \quad \quad \quad \space \space \downarrow[/latex]

[latex]\quad\text{Pressure} \times \text{Area} = \text{Force}[/latex]

Proportions