Solving Systems Of Equations By Substitution Method 2a + 3b = 16 And 2a - 4b = 2

Hey there, math enthusiasts! Today, we're diving deep into the fascinating world of solving systems of linear equations using the substitution method. We're going to tackle a classic problem: finding the values of 'a' and 'b' that satisfy both equations 2a + 3b = 16 and 2a - 4b = 2. Buckle up, because we're about to make math magic happen!

Cracking the Code: Understanding the Substitution Method

The substitution method is a powerful technique for solving systems of equations. The core idea is simple: we solve one equation for one variable and then substitute that expression into the other equation. This clever move eliminates one variable, leaving us with a single equation that we can easily solve. Once we've found the value of one variable, we can substitute it back into either of the original equations to find the value of the other variable. Think of it like a mathematical puzzle where we're strategically swapping pieces to reveal the solution.

Why is the substitution method so awesome? Well, it's incredibly versatile. It works like a charm for systems of two equations with two variables, but it can also be extended to solve more complex systems with more variables. Plus, it's a great way to build your algebraic skills and deepen your understanding of how equations work together. Now, let's get down to business and apply the substitution method to our problem.

Step-by-Step: Solving 2a + 3b = 16 and 2a - 4b = 2

Let's break down the process into manageable steps. Trust me, it's easier than it looks!

Step 1: Choose an Equation and Solve for One Variable

The first step is to pick one of the equations and solve it for one of the variables. It doesn't matter which equation or which variable you choose – the final answer will be the same. However, to make our lives easier, we'll look for the equation and variable that will result in the simplest expression. In our case, both equations have a '2a' term, so let's choose the second equation, 2a - 4b = 2, and solve it for 'a'.

To isolate 'a', we'll add 4b to both sides of the equation:

2a - 4b + 4b = 2 + 4b

This simplifies to:

2a = 2 + 4b

Now, we'll divide both sides by 2:

2a / 2 = (2 + 4b) / 2

This gives us:

a = 1 + 2b

Ta-da! We've successfully solved the second equation for 'a'. We now have an expression for 'a' in terms of 'b'. This is a crucial step in the substitution method.

Step 2: Substitute the Expression into the Other Equation

Now comes the fun part: substitution! We'll take the expression we just found for 'a' (a = 1 + 2b) and substitute it into the other equation, which is 2a + 3b = 16. Remember, we want to eliminate one variable, and by substituting, we're doing just that.

So, wherever we see 'a' in the first equation, we'll replace it with '1 + 2b'. This gives us:

2(1 + 2b) + 3b = 16

Notice that we've replaced 'a' with an entire expression. This is where the substitution method gets its name. Now, we have an equation with only one variable, 'b'. We're one step closer to cracking the code!

Step 3: Solve the Resulting Equation

Our next mission is to solve the equation we obtained in the previous step: 2(1 + 2b) + 3b = 16. This equation only involves 'b', so we can use our algebraic skills to find its value.

First, we'll distribute the 2:

2 + 4b + 3b = 16

Next, we'll combine like terms:

2 + 7b = 16

Now, we'll subtract 2 from both sides:

7b = 14

Finally, we'll divide both sides by 7:

b = 2

Yes! We've found the value of 'b'. It's equal to 2. We're halfway there! With the substitution method, we're systematically unraveling the puzzle.

Step 4: Substitute the Value Back to Find the Other Variable

We've discovered that b = 2. Now, we need to find the value of 'a'. Remember that expression we found earlier, a = 1 + 2b? This is where it comes in handy. We'll simply substitute the value of 'b' (which is 2) into this expression:

a = 1 + 2(2)

Simplifying, we get:

a = 1 + 4

a = 5

Fantastic! We've found the value of 'a'. It's equal to 5. We've successfully solved for both variables using the substitution method.

Step 5: Check Your Solution

Before we celebrate, it's always a good idea to check our solution. This helps us ensure that we haven't made any mistakes along the way. To check our solution, we'll substitute the values of 'a' and 'b' (a = 5, b = 2) into both of the original equations:

For the first equation, 2a + 3b = 16:

2(5) + 3(2) = 10 + 6 = 16

This checks out! The equation is satisfied.

For the second equation, 2a - 4b = 2:

2(5) - 4(2) = 10 - 8 = 2

This also checks out! The equation is satisfied.

Since our solution satisfies both equations, we can confidently say that a = 5 and b = 2 is the correct solution to the system of equations.

The Grand Finale: The Solution and Its Significance

We've done it! We've successfully solved the system of equations 2a + 3b = 16 and 2a - 4b = 2 using the substitution method. Our solution is a = 5 and b = 2. This means that the point (5, 2) is the intersection point of the two lines represented by these equations. In other words, it's the only point that lies on both lines.

The substitution method is not just a mathematical trick; it's a powerful tool for solving real-world problems. Systems of equations arise in various fields, such as physics, engineering, economics, and computer science. By mastering the substitution method, you're equipping yourself with a valuable problem-solving skill that can be applied in many different contexts.

Mastering the Substitution Method: Tips and Tricks

To truly become a pro at the substitution method, here are some extra tips and tricks to keep in mind:

• Choose wisely: When deciding which equation and variable to solve for in Step 1, look for the simplest option. This can save you time and effort.
• Be careful with signs: Pay close attention to positive and negative signs, especially when substituting expressions.
• Distribute correctly: Make sure to distribute any coefficients properly when substituting.
• Check your work: Always check your solution by substituting the values back into the original equations.
• Practice makes perfect: The more you practice the substitution method, the more comfortable and confident you'll become.

Level Up Your Math Skills: Exploring Other Methods

The substitution method is just one way to solve systems of equations. Another popular method is the elimination method, which involves adding or subtracting the equations to eliminate a variable. It's worth exploring both methods to see which one you prefer and which one is best suited for different types of problems. There are also graphical methods for solving systems of equations, which can provide a visual understanding of the solution.

Keep exploring, keep practicing, and keep pushing your math skills to the next level! You've got this!

Conclusion: The Power of the Substitution Method

We've journeyed through the ins and outs of the substitution method, solving a classic system of equations along the way. We've seen how this method allows us to systematically eliminate variables and find solutions. Remember, math is not just about memorizing formulas; it's about understanding concepts and developing problem-solving skills. The substitution method is a testament to the power of algebraic manipulation and logical reasoning.

So, the next time you encounter a system of equations, don't shy away! Embrace the challenge, dust off your substitution method skills, and watch as the solutions unfold. You've got the tools, the knowledge, and the determination to conquer any mathematical obstacle that comes your way. Happy solving, guys!