Solving Linear Equations Substitution Method Examples

Hey guys! Today, we're diving into the substitution method for solving pairs of linear equations. This is a super useful technique in algebra, and we're going to break it down step by step with some examples. Let's get started!

Understanding the Substitution Method

The substitution method is all about solving for one variable in terms of the other and then substituting that expression into the other equation. This simplifies the system into a single equation with one variable, which is much easier to solve. Once we find the value of one variable, we can easily find the value of the other. It might sound a bit complex now, but trust me, it's pretty straightforward once you see it in action.

Why Use Substitution?

Why bother with substitution when there are other methods like elimination or graphing? Well, the substitution method is particularly handy when one of the equations is already solved (or easily solvable) for one variable. It's also a great method to have in your toolbox because it reinforces your understanding of how variables relate to each other in a system of equations. Plus, it's a skill that comes in handy in higher-level math, so mastering it now is a solid investment in your future math prowess. So, stick with me, and let's make sure you nail this method!

Example 1: Solving the First Pair of Equations

Let's tackle our first pair of equations:

$$\begin{array}{l} 2x = 5y + 12 \\ 3x - 2y + 4 = 0 \end{array}$$

Step 1: Isolate One Variable

The first step in the substitution method is to isolate one variable in one of the equations. Looking at our equations, the first one, 2x = 5y + 12, seems easier to work with for isolating x. We can do this by dividing both sides by 2:

$$x = \frac{5y + 12}{2}$$

Now we have x expressed in terms of y. This is our key to unlocking the solution!

Step 2: Substitute

Next, we substitute this expression for x into the second equation, which is 3x - 2y + 4 = 0. Replacing x with (5y + 12) / 2, we get:

$$3\left(\frac{5y + 12}{2}\right) - 2y + 4 = 0$$

This looks a bit messy, but don't worry! We're going to simplify it. By substituting, we've eliminated x from the second equation, and we're left with an equation that only involves y. This is exactly what we wanted!

Step 3: Simplify and Solve for y

Now, let's simplify the equation and solve for y. First, distribute the 3:

$$\frac{15y + 36}{2} - 2y + 4 = 0$$

To get rid of the fraction, multiply the entire equation by 2:

$$15y + 36 - 4y + 8 = 0$$

Combine like terms:

$$11y + 44 = 0$$

Subtract 44 from both sides:

$$11y = -44$$

Finally, divide by 11:

$$y = -4$$

Great! We've found the value of y. Now we're halfway there!

Step 4: Substitute y Back to Find x

We now know that y = -4. To find the value of x, we substitute this value back into the expression we found in Step 1: x = (5y + 12) / 2.

Plugging in y = -4, we get:

$$x = \frac{5(-4) + 12}{2}$$

$$x = \frac{-20 + 12}{2}$$

$$x = \frac{-8}{2}$$

$$x = -4$$

So, x = -4. We've found both x and y!

Step 5: Write the Solution

Our solution is the pair of values x = -4 and y = -4. We can write this as an ordered pair: (-4, -4). This means that the point where the two lines represented by our original equations intersect is (-4, -4). And that's how you solve a pair of linear equations using substitution!

Example 2: Solving the Second Pair of Equations

Okay, let's move on to the second pair of equations and reinforce our understanding of the substitution method. The equations are:

$$\begin{array}{l} x + y = 7 \\ 2x - 3y = 11 \end{array}$$

Step 1: Isolate One Variable

In this set of equations, the first equation, x + y = 7, looks super simple to manipulate. Let's isolate x by subtracting y from both sides:

$$x = 7 - y$$

We've now expressed x in terms of y. This is a great start!

Step 2: Substitute

Next, we substitute this expression for x into the second equation, which is 2x - 3y = 11. Replacing x with (7 - y), we get:

$$2(7 - y) - 3y = 11$$

By substituting, we've eliminated x from the second equation, and we're left with an equation that only involves y. Just like before, this makes it much easier to solve!

Step 3: Simplify and Solve for y

Let's simplify the equation and solve for y. First, distribute the 2:

$$14 - 2y - 3y = 11$$

Combine like terms:

$$14 - 5y = 11$$

Subtract 14 from both sides:

$$-5y = -3$$

Finally, divide by -5:

$$y = \frac{3}{5}$$

Alright! We've found the value of y. It's a fraction, but don't let that scare you. We'll work with it.

Step 4: Substitute y Back to Find x

We now know that y = 3/5. To find the value of x, we substitute this value back into the expression we found in Step 1: x = 7 - y.

Plugging in y = 3/5, we get:

$$x = 7 - \frac{3}{5}$$

To subtract the fraction, we need a common denominator. Let's rewrite 7 as 35/5:

$$x = \frac{35}{5} - \frac{3}{5}$$

$$x = \frac{32}{5}$$

So, x = 32/5. We've found both x and y! Even though they're fractions, the process is exactly the same.

Step 5: Write the Solution

Our solution is the pair of values x = 32/5 and y = 3/5. We can write this as an ordered pair: (32/5, 3/5). This is the point where the two lines represented by our original equations intersect. See? We nailed it!

Tips and Tricks for the Substitution Method

Before we wrap up, let's go over some tips and tricks to make your substitution method skills even stronger:

1. Choose Wisely: When deciding which variable to isolate, look for the equation where a variable has a coefficient of 1 or -1. This will make the isolation step much easier.
2. Double-Check Your Work: Substitution can involve some messy algebra, so always double-check your work, especially when dealing with fractions or negative signs. A small mistake can throw off your entire solution.
3. Stay Organized: Keep your work neat and organized. Write each step clearly, and make sure you're substituting into the correct equation. This will help prevent errors and make it easier to track your progress.
4. Verify Your Solution: After finding x and y, plug them back into the original equations to make sure they satisfy both. This is a great way to catch any mistakes and ensure your solution is correct.
5. Practice, Practice, Practice: Like any math skill, mastering the substitution method takes practice. Work through plenty of examples, and don't be afraid to ask for help if you get stuck. The more you practice, the more confident you'll become.

Conclusion

So, there you have it! The substitution method for solving pairs of linear equations. We've covered the steps, worked through examples, and shared some tips to help you master this technique. Remember, the key is to isolate one variable, substitute, simplify, and solve. With practice, you'll be solving these equations like a pro! Keep up the great work, and I'll catch you in the next math adventure!