Solving three-step linear equations is a fundamental skill in algebra that involves isolating the variable on one side of the equation. This technique is crucial for solving various mathematical problems, scientific equations, and real-world scenarios. Understanding the principles and steps involved in solving three-step linear equations empower individuals to tackle more complex equations and advance their analytical abilities.
To effectively solve three-step linear equations, it’s essential to follow a systematic approach. The first step entails isolating the variable term on one side of the equation. This can be achieved by performing inverse operations, such as adding or subtracting the same value from both sides of the equation. The goal is to simplify the equation and eliminate any constants or coefficients that are attached to the variable.
Once the variable term is isolated, the next step involves solving for the variable. This typically involves dividing both sides of the equation by the coefficient of the variable. By performing this operation, we effectively isolate the variable and determine its value. It’s important to note that dividing by zero is undefined, so caution must be exercised when dealing with equations that involve zero as the coefficient of the variable.
Understanding the Concept of a Three-Step Linear Equation
A three-step linear equation is an algebraic equation that can be solved in three basic steps. It typically has the form ax + b = c, where a, b, and c are numerical coefficients that can be positive, negative, or zero.
To understand the concept of a three-step linear equation, it’s crucial to grasp the following key ideas:
Isolating the Variable (x)
The goal of solving a three-step linear equation is to isolate the variable x on one side of the equation and express it in terms of a, b, and c. This isolation process involves performing a series of mathematical operations while maintaining the equality of the equation.
The three basic steps involved in solving a linear equation are summarized in the table below:
| Step | Operation | Purpose |
|---|---|---|
| 1 | Isolate the variable term (ax) on one side of the equation. | Remove or add any constant terms (b) to both sides of the equation to isolate the variable term. |
| 2 | Simplify the equation by dividing or multiplying by the coefficient of the variable (a). | Isolate the variable (x) on one side of the equation by dividing or multiplying both sides by a, which is the coefficient of the variable. |
| 3 | Solve for the variable (x) by simplifying the remaining expression. | Perform any necessary arithmetic operations to find the numerical value of the variable. |
Simplifying the Equation with Addition or Subtraction
The second step in solving a three-step linear equation involves simplifying the equation by adding or subtracting the same value from both sides of the equation. This process does not alter the solution to the equation because adding or subtracting the same value from both sides of an equation preserves the equality.
There are two scenarios to consider when simplifying an equation using addition or subtraction:
| Scenario | Operation |
|---|---|
| When the variable is added to (or subtracted from) both sides of the equation | Subtract (or add) the variable from both sides |
| When the variable has a coefficient other than 1 added to (or subtracted from) both sides of the equation | Divide both sides by the coefficient of the variable |
For example, let’s consider the equation:
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2x + 5 = 13
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In this equation, 5 is added to both sides of the equation:
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2x + 5 – 5 = 13 – 5
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Simplifying the equation, we get:
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2x = 8
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Now, to solve for x, we divide both sides by 2:
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(2x) / 2 = 8 / 2
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Simplifying the equation, we find the value of x:
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x = 4
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Combining Like Terms
Combining like terms is the process of adding or subtracting terms with the same variable and exponent. To combine like terms, simply add or subtract the coefficients (the numbers in front of the variables) and keep the same variable and exponent. For example:
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3x + 2x = 5x
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In this example, we have two like terms, 3x and 2x. We can combine them by adding their coefficients to get 5x.
Isolating the Variable
Isolating the variable is the process of getting the variable by itself on one side of the equation. To isolate the variable, we need to undo any operations that have been done to it. Here is a step-by-step guide to isolating the variable:
- If the variable is being added to or subtracted from a constant, subtract or add the constant to both sides of the equation.
- If the variable is being multiplied or divided by a constant, divide or multiply both sides of the equation by the constant.
- Repeat steps 1 and 2 until the variable is isolated on one side of the equation.
For example, let’s isolate the variable in the equation:
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3x – 5 = 10
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- Add 5 to both sides of the equation to get:
- Divide both sides of the equation by 3 to get:
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3x = 15
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x = 5
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Therefore, the solution to the equation is x = 5.
| Step | Equation |
|---|---|
| 1 | 3x – 5 = 10 |
| 2 | 3x = 15 |
| 3 | x = 5 |
Using Multiplication or Division to Isolate the Variable
In cases where the variable is multiplied or divided by a coefficient, you can undo the operation by performing the opposite operation on both sides of the equation. This will isolate the variable on one side of the equation and allow you to solve for its value.
Multiplication
If the variable is multiplied by a coefficient, divide both sides of the equation by the coefficient to isolate the variable.
Example: Solve for x in the equation 3x = 15.
| Step | Equation |
|---|---|
| 1 | Divide both sides by 3 |
| 2 | x = 5 |
Division
If the variable is divided by a coefficient, multiply both sides of the equation by the coefficient to isolate the variable.
Example: Solve for y in the equation y/4 = 10.
| Step | Equation |
|---|---|
| 1 | Multiply both sides by 4 |
| 2 | y = 40 |
By performing multiplication or division to isolate the variable, you effectively undo the operation that was performed on the variable originally. This allows you to solve for the value of the variable directly.
Verifying the Solution through Substitution
Once you have found a potential solution to your three-step linear equation, it’s crucial to verify its accuracy. Substitution is a simple yet effective method for doing so. To verify the solution:
1. Substitute the potential solution into the original equation: Replace the variable in the equation with the value you found as the solution.
2. Simplify the equation: Perform the necessary mathematical operations to simplify the left-hand side (LHS) and right-hand side (RHS) of the equation.
3. Check for equality: If the LHS and RHS of the simplified equation are equal, then the potential solution is indeed a valid solution to the original equation.
4. If the equation is not equal: If the LHS and RHS of the simplified equation do not match, then the potential solution is incorrect, and you need to repeat the steps to find the correct solution.
Example:
Consider the following equation: 2x + 5 = 13.
Let’s say you have found the potential solution x = 4. To verify it:
| Step | Action |
|---|---|
| 1 | Substitute x = 4 into the equation: 2(4) + 5 = 13 |
| 2 | Simplify the equation: 8 + 5 = 13 |
| 3 | Check for equality: The LHS and RHS are equal (13 = 13), so the potential solution is valid. |
Simplifying the Equation by Combining Fractions
When you encounter fractions in your equation, it can be helpful to combine them for easier manipulation. Here are some steps to do so:
1. Find a Common Denominator
Look for the Least Common Multiple (LCM) of the denominators of the fractions. This will become your new denominator.
2. Multiply Numerators and Denominators
Once you have the LCM, multiply both the numerator and denominator of each fraction by the LCM divided by the original denominator. This will give you equivalent fractions with the same denominator.
3. Add or Subtract Numerators
If the fractions have the same sign (both positive or both negative), simply add the numerators and keep the original denominator. If they have different signs, subtract the smaller numerator from the larger and make the resulting numerator negative.
For example:
| Original Equation: | 3/4 – 1/6 |
| LCM of 4 and 6: | 12 |
| Equivalent Fractions: | 9/12 – 2/12 |
| Simplified Equation: | 7/12 |
Dealing with Equations Involving Decimal Coefficients
When dealing with decimal coefficients, it is essential to be cautious and accurate. Here’s a detailed guide to help you solve equations involving decimal coefficients:
Step 1: Convert the Decimal to a Fraction
Begin by converting the decimal coefficients into their equivalent fractions. This can be done by multiplying the decimal by 10, 100, or 1000, as many times as the number of decimal places. For example, 0.25 can be converted to 25/100, 0.07 can be converted to 7/100, and so on.
Step 2: Simplify the Fractions
Once you have converted the decimal coefficients to fractions, simplify them as much as possible. This involves finding the greatest common divisor (GCD) of the numerator and denominator and dividing both by the GCD. For example, 25/100 can be simplified to 1/4.
Step 3: Clear the Denominators
To clear the denominators, multiply both sides of the equation by the least common multiple (LCM) of the denominators. This will eliminate the fractions and make the equation easier to solve.
Step 4: Solve the Equation
Once the denominators have been cleared, the equation becomes a simple linear equation that can be solved using the standard algebraic methods. This may involve addition, subtraction, multiplication, or division.
Step 5: Check Your Answer
After solving the equation, check your answer by substituting it back into the original equation. If both sides of the equation are equal, then your answer is correct.
Example:
Solve the equation: 0.25x + 0.07 = 0.52
1. Convert the decimal coefficients to fractions:
0.25 = 25/100 = 1/4
0.07 = 7/100
0.52 = 52/100
2. Simplify the fractions:
1/4
7/100
52/100
3. Clear the denominators:
4 * (1/4x + 7/100) = 4 * (52/100)
x + 7/25 = 26/25
4. Solve the equation:
x = 26/25 – 7/25
x = 19/25
5. Check your answer:
0.25 * (19/25) + 0.07 = 0.52
19/100 + 7/100 = 52/100
26/100 = 52/100
0.52 = 0.52
Handling Equations with Negative Coefficients or Constants
When dealing with negative coefficients or constants in a three-step linear equation, extra care is required to maintain the integrity of the equation while isolating the variable.
For example, consider the equation:
-2x + 5 = 11
To isolate x on one side of the equation, we need to first eliminate the constant term (5) on that side. This can be done by subtracting 5 from both sides, as shown below:
-2x + 5 – 5 = 11 – 5
-2x = 6
Next, we need to eliminate the coefficient of x (-2). We can do this by dividing both sides by -2, as shown below:
-2x/-2 = 6/-2
x = -3
Therefore, the solution to the equation -2x + 5 = 11 is x = -3.
It’s important to note that when multiplying or dividing by a negative number, the signs of the other terms in the equation may change. To ensure accuracy, it’s always a good idea to check your solution by substituting it back into the original equation.
To summarize, the steps involved in handling negative coefficients or constants in a three-step linear equation are as follows:
| Step | Description |
|---|---|
| 1 | Eliminate the constant term by adding or subtracting the same number from both sides of the equation. |
| 2 | Eliminate the coefficient of the variable by multiplying or dividing both sides of the equation by the reciprocal of the coefficient. |
| 3 | Check your solution by substituting it back into the original equation. |
Solving Equations with Parentheses or Brackets
When an equation contains parentheses or brackets, it’s crucial to follow the order of operations. First, simplify the expression inside the parentheses or brackets to a single value. Then, substitute this value back into the original equation and solve as usual.
Example:
Solve for x:
2(x – 3) + 5 = 11
Step 1: Simplify the Expression in Parentheses
2(x – 3) = 2x – 6
Step 2: Substitute the Simplified Expression
2x – 6 + 5 = 11
Step 3: Solve the Equation
2x – 1 = 11
2x = 12
x = 6
Therefore, x = 6 is the solution to the equation.
Table of Examples:
| Equation | Solution |
|---|---|
| 2(x + 1) – 3 = 5 | x = 2 |
| 3(2x – 5) + 1 = 16 | x = 3 |
| (x – 2)(x + 3) = 0 | x = 2 or x = -3 |
Real-World Applications of Solving Three-Step Linear Equations
Solving three-step linear equations has numerous practical applications in real-world scenarios. Here’s a detailed exploration of its uses in various fields:
1. Finance
Solving three-step linear equations allows us to calculate loan payments, interest rates, and investment returns. For example, determining the monthly payments for a home loan requires solving an equation relating the loan amount, interest rate, and loan term.
2. Physics
In physics, understanding motion and kinematics involves solving linear equations. Equations like v = u + at, where v represents the final velocity, u represents the initial velocity, a represents acceleration, and t represents time, help us analyze motion under constant acceleration.
3. Chemistry
Chemical reactions and stoichiometry rely on solving three-step linear equations. They help determine concentrations, molar masses, and reaction yields based on chemical equations and mass-to-mass relationships.
4. Engineering
From structural design to fluid dynamics, engineers frequently employ three-step linear equations to solve real-world problems. They calculate forces, pressures, and flow rates using equations involving variables such as area, density, and velocity.
5. Medicine
In medicine, dosage calculations require solving three-step linear equations. Determining the appropriate dose of medication based on a patient’s weight, age, and medical condition involves solving equations to ensure safe and effective treatment.
6. Economics
Economic models use linear equations to analyze demand, supply, and market equilibrium. They can determine equilibrium prices, quantity demanded, and consumer surplus by solving these equations.
7. Transportation
In transportation, equations involving distance, speed, and time are used to calculate arrival times, fuel consumption, and average speeds. Solving these equations helps optimize routes and schedules.
8. Biology
Population growth models often use three-step linear equations. Equations like y = mx + b, where y represents population size, m represents growth rate, x represents time, and b represents the initial population, help predict population dynamics.
9. Business
Businesses use linear equations to model revenue, profit, and cost functions. They can determine break-even points, optimize pricing strategies, and forecast financial outcomes by solving these equations.
10. Data Analysis
In data analysis, linear regression is a common technique for modeling relationships between variables. It involves solving a three-step linear equation to find the best-fit line and extract insights from data.
| Industry | Application |
|---|---|
| Finance | Loan payments, interest rates, investment returns |
| Physics | Motion and kinematics |
| Chemistry | Chemical reactions, stoichiometry |
| Engineering | Structural design, fluid dynamics |
| Medicine | Dosage calculations |
| Economics | Demand, supply, market equilibrium |
| Transportation | Arrival times, fuel consumption, average speeds |
| Biology | Population growth models |
| Business | Revenue, profit, cost functions |
| Data Analysis | Linear regression |
How To Solve A Three Step Linear Equation
Solving a three-step linear equation involves isolating the variable (usually represented by x) on one side of the equation and the constant on the other side. Here are the steps to solve a three-step linear equation:
- Step 1: Simplify both sides of the equation. This may involve combining like terms and performing basic arithmetic operations such as addition or subtraction.
- Step 2: Isolate the variable term. To do this, perform the opposite operation on both sides of the equation that is next to the variable. For example, if the variable is subtracted from one side, add it to both sides.
- Step 3: Solve for the variable. Divide both sides of the equation by the coefficient of the variable (the number in front of it). This will give you the value of the variable.
People Also Ask
How do you check your answer for a three-step linear equation?
To check your answer, substitute the value you found for the variable back into the original equation. If both sides of the equation are equal, then your answer is correct.
What are some examples of three-step linear equations?
Here are some examples of three-step linear equations:
- 3x + 5 = 14
- 2x – 7 = 3
- 5x + 2 = -3
Can I use a calculator to solve a three-step linear equation?
Yes, you can use a calculator to solve a three-step linear equation. However, it is important to understand the steps involved in solving the equation so that you can check your answer and troubleshoot any errors.
