How to Solve Linear Equations with Variables on Both Sides

Solving linear equations can seem hard at first, especially for Year 10 students learning algebra. But don’t worry! There are some easy steps that can help you find the solution more clearly. Let’s break it down together.

Step 1: Move All Variables to One Side

First, let's get all the variable terms on one side of the equation. We can do this by adding or subtracting.

For example, take this equation:

$$3x + 5 = 2x + 12.$$

To move the ( 2x ) to the left side, we subtract ( 2x ) from both sides:

$$3x - 2x + 5 = 2x - 2x + 12,$$

which simplifies to:

$$x + 5 = 12.$$

Now we see just one variable, ( x ), which is much easier to handle!

Step 2: Isolate the Variable

Next, we want to isolate ( x ). Once we have an equation like ( x + a = b ), where ( a ) is a constant, we can subtract ( a ) from both sides to find ( x ):

$$x + 5 - 5 = 12 - 5,$$

giving us:

$$x = 7.$$

This two-step process—moving the variables and then isolating them—is really important!

Step 3: Combine Like Terms

Sometimes, it's useful to combine like terms before isolating the variable. If both sides have similar terms, simplifying those first can help.

For instance, look at this equation:

$$2x + 3 = 4 + x + 2.$$

Before moving terms, we can combine the numbers on the right:

$$2x + 3 = x + 6.$$

Now, we move the ( x ) terms over by subtracting ( x ) from both sides:

$$2x - x + 3 = 6,$$

simplifying to:

$$x + 3 = 6.$$

From here, we can isolate ( x ):

$$x + 3 - 3 = 6 - 3,$$

which results in:

$$x = 3.$$

Step 4: Align and Rearrange

For more complex equations, it can help to rearrange them first. This makes it easier to see what steps to take next.

Let's look at this equation:

$$5x - 7 = 3x + 9 - 2x.$$

First, simplify the right side:

$$5x - 7 = 3x + 9 - 2x \implies 5x - 7 = (3x - 2x) + 9 \implies 5x - 7 = x + 9.$$

Next, get the ( x ) terms together:

$$5x - x - 7 = 9,$$

which simplifies to:

$$4x - 7 = 9.$$

Now, add 7 to both sides:

$$4x = 9 + 7 \implies 4x = 16.$$

Finally, divide by 4:

$$x = 4.$$

Step 5: Check Your Work

Always remember to check your answer. Once you find a solution, put that value back into the original equation to see if it works.

For our last example, we check:

$$5(4) - 7 \stackrel{?}{=} 3(4) + 9 - 2(4) \implies 20 - 7 \stackrel{?}{=} 12 + 9 - 8 \implies 13 = 13,$$

which means our solution is correct!

Step 6: Collecting Like Terms

Knowing how to collect like terms helps us simplify equations faster. For example:

$$2(3x - 4) = 6 + 3(2x - 1).$$

By distributing, we get:

$$6x - 8 = 6 + 6x - 3.$$

This simplifies to:

$$6x - 8 = 3 + 6x.$$

Now we see the ( 6x ) terms will cancel each other out if we subtract ( 6x ) from both sides:

$$6x - 6x - 8 = 3 + 6x - 6x \implies -8 = 3.$$

This means there’s no solution for this equation, which is just as important to know!

Step 7: Using the Distributive Property

Don’t forget the distributive property, too! This helps when an equation has parentheses. For instance:

$$4(x + 2) = 3(2x - 1).$$

After distributing, we have:

$$4x + 8 = 6x - 3.$$

Now, we can easily isolate ( x ):

$$4x - 6x = -3 - 8 \implies -2x = -11 \implies x = \frac{11}{2} \text{ or } 5.5.$$

In Conclusion

Solving equations with variables on both sides doesn’t have to be scary! By following the steps like moving variables, combining like terms, checking for no solutions, and using the distributive property, you can become really good at it. With practice, these methods will help you feel more confident in your math skills!