how to solve system of equations by substitution

How to Solve System of Equations by Substitution: A Step-by-Step Guide for Success 💡

Solving systems of equations can seem daunting, but you got this! 😊 Let’s dive into a step-by-step guide that will empower you to conquer this math hurdle. Whether you’re a seasoned pro or just starting your math journey, this article will break down the substitution method and equip you with the tools to tackle equations confidently. Get ready to simplify equations and uncover those hidden solutions! 🎉

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What is a System of Equations?

A system of equations is like a detective game where you’re on the hunt for the values of unknown variables. Each equation is a clue leading you closer to the solution, and the final answer is the moment when the puzzle pieces fall into place. 👌

Solving by Substitution: A Step-by-Step Guide

Let’s get to the nitty-gritty of solving systems of equations by substitution. Follow these steps, and you’ll be unstoppable:

1. Choose an Equation with an Isolated Variable:

Identify an equation where one variable is by itself on one side of the equation. This will be your starting point.

2. Solve for the Variable:

Solve the chosen equation for the variable you isolated in step 1. This will give you an expression for that variable.

3. Substitute into the Other Equation:

Take the expression you found in step 2 and plug it into the other equation. This step replaces the variable with its equivalent expression.

4. Solve for the Remaining Variable:

Solve the new equation you created in step 3 for the remaining variable. This will give you the value of that variable.

5. Substitute Back into the Original Equation:

Take the value you found in step 4 and substitute it back into the original equation. This will give you the value of the first variable.

6. Check Your Solution:

Plug your values for both variables back into both original equations to ensure they satisfy both equations. If they do, you’ve cracked the code! 🥳

Why Substitution?

Why use substitution instead of other methods? Well, my friend, substitution shines when one of the equations is already solved for a variable or when one variable is easily isolated. It’s a straightforward approach that can save you precious time and gray hair. 😉

Examples for Clarity

Let’s try out our newfound substitution skills with some examples:

Example 1:

Solve the system of equations:

x + y = 5
x - y = 1

Solution:

• Step 1: Isolating y in the second equation, we get y = x – 1.
• Step 2: Substituting this expression into the first equation, we get x + (x – 1) = 5.
• Step 3: Solving the new equation, we get 2x = 6, which gives us x = 3.
• Step 4: Substituting x = 3 back into y = x – 1, we get y = 2.
• Step 5: Our solution is (x, y) = (3, 2).

Example 2:

Solve the system of equations:

2x + 3y = 11
x - y = 4

Solution:

• Step 1: Isolating y in the second equation, we get y = x – 4.
• Step 2: Substituting this expression into the first equation, we get 2x + 3(x – 4) = 11.
• Step 3: Solving the new equation, we get 5x – 12 = 11, which gives us x = 4.7.
• Step 4: Substituting x = 4.7 back into y = x – 4, we get y = 0.7.
• Step 5: Our solution is (x, y) = (4.7, 0.7).

Comparison Table: Substitution vs. Other Methods

Tips for Success

• Simplify First: Before substituting, simplify both equations to make the process easier.
• Choose Wisely: Select the equation that gives you the most straightforward expression when isolating a variable.
• Check Your Work: Always plug your solution back into both original equations to verify its accuracy.
• Don’t Fear Fractions: In some cases, substitution may result in fractions or decimals as solutions. Embrace them! 😉
• Practice Makes Perfect: The more you practice, the more confident you’ll become in solving systems of equations. 💪

Conclusion

Well, there you have it! You’re now fully equipped to tackle systems of equations with confidence using the substitution method. Remember, practice is key, so grab some equations and give it a go. And hey, if you want to expand your math prowess, check out our other articles on solving equations. Keep conquering those math challenges, and don’t forget to have some fun along the way! 🎉

FAQ about Solving Systems by Substitution

### 1. What is substitution?
Answer: Substitution is a method to solve a system of equations by replacing one variable with an equivalent expression involving the other variable.

### 2. How to choose which equation to solve for one variable first?
Answer: Choose the equation with the variable that appears by itself (or with a coefficient of 1) in at least one term.

### 3. How to isolate the variable in the equation?
Answer: Isolate the variable by adding or subtracting the same value on both sides of the equation, or by multiplying or dividing both sides by the same non-zero value (P-A-S).

### 4. How to substitute the isolated variable into the other equation?
Answer: Replace the variable in the other equation with the expression you obtained by isolating it (P-A-S).

### 5. How to solve for the remaining variable in the substituted equation?
Answer: Use the steps of isolating the variable in the substituted equation to find the value for the remaining variable (P-A-S).

### 6. How to check if the solution is correct?
Answer: Substitute the values of the variables back into both original equations. They should both be true if the solution is correct.

### 7. What to do if the substitution result is a contradiction (e.g., 5=2)?
Answer: The system has no solution in this case.

### 8. What to do if the substitution result is an identity (e.g., 0=0)?
Answer: The system has infinitely many solutions in this case.

### 9. How to use a table to solve a system using substitution?
Answer: Create a table with columns for each variable, isolate one variable in one equation, substitute into the second equation to find the other variable, and continue filling out the table until a solution is found.

### 10. What are some common mistakes to avoid when using substitution?
Answer: Making algebraic errors, forgetting to check the solution, and not considering special cases (no solution or infinite solutions).