Problem 29 - Algebra 1: Concepts and Skills

Question

Solve the equation. $$ 10(2 x+4)=-(-8-9 x)+3 x $$

Step-by-Step Solution

Answer

The solution to the given equation is \( x = \frac{8}{3} \)

Step 1: Expand both sides of the equation

Apply the distributive property of multiplication over addition or subtraction on both sides of the equation. This results in: \( 20x + 40 = -(-8 - 9x) + 3x \)

Step 2: Simplify by combining like terms

Combine like terms on each side of the equation and simplify the equation wherever possible. Now the equation simplifies to: \( 20x + 40 = 8 + 9x + 3x \)

Step 3: Group the variable x on one side

To isolate the variable x, you subtract 20x from both sides (which is the coefficient of x on the left hand side). The equation after this operation is: \( 40 = 8 + 12x \)

Step 4: Further Simplification of the Equation

Continue simplifying the equation and solving for x. This implies subtracting 8 from both sides of the equation, which results in: \( 32 = 12x \)

Step 5: Solve for the Variable x

Now that x is alone on one side, finalize the equation by solving for x. This result can be achieved by dividing both sides of the equation by 12, resulting in the final equation: \( x = \frac{32}{12} \)

Key Concepts

Distributive Property

Grasping the distributive property is a foundational skill in algebra. It's a way to multiply a single term by a group of terms within parentheses. In essence, you distribute the multiplication to each term inside. For example, if you have an equation like 3(a + b), applying the distributive property involves multiplying 3 both by a and by b, resulting in 3a + 3b.

Consider our sample problem 10(2x + 4). Using distributive property, we multiply both 2x and 4 by 10, simplifying the equation to 20x + 40. This property is essential to solve linear equations, as it allows you to combine like terms, which is our next concept.

Combining Like Terms

Now, let's demystify the concept of combining like terms. This is a crucial step in simplifying algebraic expressions. Like terms are terms that have identical variable parts, such as 2x and 3x or 5 and -2 (which are both constants)

Why Combine Like Terms?

In our exercise, after distributing, we got 20x + 40 = 8 + 9x + 3x. Notice the 9x and 3x on the right side of the equal sign. They are 'like terms', as they both have the same variable x. To simplify, they are added together to get 12x, bringing us to 20x + 40 = 8 + 12x. Combining like terms makes the equation easier to solve, which sets the stage for isolating the variable.

Isolate the Variable

Isolating the variable is your final destination on the journey to solving a linear equation. To 'isolate' means to get the variable on one side of the equation by itself. From the equation 20x + 40 = 8 + 12x, we need to get x on its own on one side. This is done by performing the same operation on both sides of the equation.

Step-by-Step Isolation

First, let's subtract 12x from both sides which gives us 8x + 40 = 8. Next, we subtract 40 from both sides which leads us to 8x = -32. Finally, divide both sides by 8 to get x completely isolated: x = -4. The key here is to perform the arithmetic carefully, ensuring you're consistently applying operations across the entire equation to maintain balance.