Solving Systems Of Equations By Substitution A Step-by-Step Guide

Hey guys! Today, we're diving into the world of systems of equations and tackling them with a powerful technique called substitution. It might sound intimidating, but trust me, it's a super useful tool to have in your math arsenal. We're going to break down the process step-by-step, using a real example to make sure you've got it down pat. So, grab your pencils, and let's get started!

The Substitution Method: What's the Big Idea?

So, what's the deal with substitution anyway? Well, when you're faced with a system of equations – that's just a fancy way of saying you have two or more equations with the same variables – you're essentially trying to find the values for those variables that make all the equations true at the same time. Think of it like finding the perfect puzzle pieces that fit together perfectly in multiple puzzles.

The substitution method is all about isolating one variable in one equation and then "substituting" that expression into the other equation. This eliminates one variable, leaving you with a single equation that you can easily solve. Then, you can plug that value back into either of the original equations to find the value of the other variable. It's like a clever chain reaction that leads you to the solution!

Here's a breakdown of the general steps involved:

1. Choose an equation and solve for one variable. Look for an equation where one of the variables has a coefficient of 1 or -1. This makes it easier to isolate that variable without dealing with fractions. You're essentially rewriting the equation to express one variable in terms of the other.
2. Substitute the expression into the other equation. This is the heart of the method! Take the expression you found in step 1 and plug it into the other equation in place of the variable you solved for. This will give you a new equation with only one variable.
3. Solve the new equation. Now you have a simple equation with one unknown. Use your algebra skills to solve for the remaining variable.
4. Substitute the value back into either original equation. Once you've found the value of one variable, plug it back into either of the original equations to solve for the other variable. It doesn't matter which equation you choose – you'll get the same answer either way!
5. Check your solution. To make sure you've nailed it, plug both values into both original equations. If both equations are true, congratulations! You've found the solution.

Example Time: Let's Solve a System!

Okay, enough theory! Let's put the substitution method into action with the system of equations we're tackling today:

$$\begin{array}{l} 4 x-y=3 \\ 7 x-9 y=-2 \end{array}$$

Step 1: Choose an equation and solve for one variable.

Looking at the two equations, the first one (4x - y = 3) seems like a good candidate because the coefficient of y is -1. This makes it easy to isolate y. Let's do it:

4x - y = 3

Subtract 4x from both sides:

-y = 3 - 4x

Multiply both sides by -1:

y = 4x - 3

Great! We've solved the first equation for y. Now we know that y is equal to the expression 4x - 3.

Step 2: Substitute the expression into the other equation.

Now we take that expression (4x - 3) and substitute it for y in the second equation (7x - 9y = -2):

7x - 9(4x - 3) = -2

Notice how we've replaced y with the entire expression 4x - 3. It's crucial to use parentheses here to ensure you distribute the -9 correctly.

Step 3: Solve the new equation.

Now we have an equation with only one variable, x. Let's simplify and solve:

7x - 9(4x - 3) = -2

Distribute the -9:

7x - 36x + 27 = -2

Combine like terms:

-29x + 27 = -2

Subtract 27 from both sides:

-29x = -29

Divide both sides by -29:

x = 1

Awesome! We've found that x = 1.

Step 4: Substitute the value back into either original equation.

Now that we know x = 1, we can plug it back into either of the original equations to find y. Let's use the first equation (4x - y = 3) because it looks a little simpler:

4(1) - y = 3

Simplify:

4 - y = 3

Subtract 4 from both sides:

-y = -1

Multiply both sides by -1:

y = 1

Excellent! We've found that y = 1.

Step 5: Check your solution.

To be absolutely sure we've got the right answer, let's plug x = 1 and y = 1 into both original equations:

• Equation 1: 4(1) - 1 = 3 --> 4 - 1 = 3 --> 3 = 3 (True!)
• Equation 2: 7(1) - 9(1) = -2 --> 7 - 9 = -2 --> -2 = -2 (True!)

Both equations are true, so we've nailed it! Our solution is x = 1 and y = 1.

The Answer

Looking back at the answer choices, we see that the solution (1, 1) corresponds to option A. (1,1).

Key Takeaways and Tips for Success

• Practice makes perfect. The more you use the substitution method, the more comfortable you'll become with it. Work through plenty of examples to solidify your understanding.
• Choose wisely. When deciding which variable to isolate, look for the equation where a variable has a coefficient of 1 or -1. This will save you from dealing with fractions.
• Pay attention to signs. Be extra careful when distributing negative signs, especially when substituting expressions with multiple terms.
• Check your work. Always plug your solution back into both original equations to make sure it's correct. This is a crucial step to avoid careless errors.
• Stay organized. Keep your work neat and organized. This will help you avoid mistakes and make it easier to track your steps.

Mastering the Substitution Method: Why It Matters

The substitution method is more than just a trick for solving equations; it's a fundamental concept in algebra and beyond. It's used in various mathematical fields, including calculus, linear algebra, and differential equations. Understanding this method will give you a solid foundation for tackling more advanced math problems.

Moreover, the ability to solve systems of equations has practical applications in real-world scenarios. From modeling supply and demand in economics to determining the optimal mix of ingredients in a recipe, systems of equations help us solve problems in various fields.

So, take the time to master the substitution method. It's an investment in your mathematical skills that will pay off in the long run.

Conclusion: You've Got This!

And there you have it, guys! We've walked through the substitution method step-by-step, conquered a system of equations, and learned why this technique is so valuable. Remember, practice is key, so keep working at it, and you'll be solving systems of equations like a pro in no time. Happy solving!