Problem 25 - Algebra 1

Question

Solve the equation. $$8 x-3 x=10$$

Step-by-Step Solution

Answer

The solution for the equation is \(x = 2\).

Step 1: Combine Like Terms

Both terms on the left side of the equation involve the same variable \(x\). They can be combined using the properties of arithmetic: \(8x - 3x = 5x\). The equation then simplifies to \(5x = 10\).

Step 2: Isolate the Variable

To solve for \(x\), one should isolate \(x\) on one side of the equation. This can be accomplished by dividing each side of the equation by 5. \(x =\frac{10}{5}\).

Step 3: Final Solution

After dividing, it is found that \(x = 2\). This is the solution to the provided equation.

Key Concepts

Combining Like Terms

Understanding how to combine like terms is crucial when solving linear equations. Like terms are terms that contain the same variables raised to the same power. For instance, in the equation \( 8x - 3x = 10 \), both terms on the left side of the equation have the variable \(x\) without any exponents, which means they are like terms.

To combine them, simply add or subtract the coefficients (the numbers in front of the variables) following the sign that each term carries. In this example, \(8 - 3\) gives us \(5\), and since the variable is the same (\(x\)), we combine the coefficients to end up with \(5x\). This process simplifies the equation, making the subsequent steps easier.

Isolating the Variable

Once you've combined like terms, the next step in solving linear equations is to isolate the variable. This means you want to get the variable by itself on one side of the equation, with everything else on the other side.

In the simplified equation \(5x = 10\), the variable \(x\) is not yet by itself; it's multiplied by \(5\). To get \(x\) alone, we need to eliminate that \(5\) by performing the opposite operation on both sides of the equation—since \(x\) is being multiplied by \(5\), we'll divide both sides by \(5\). When we do that, we're left with \(x = \frac{10}{5}\), which simplifies to \(x = 2\). This process of isolation allows us to find the value of the variable.

Properties of Arithmetic

The properties of arithmetic make solving equations and combining like terms possible. These properties, namely the associative, commutative, and distributive properties, are fundamentally rules that describe how numbers behave.

In the context of our equation, the commutative property says that the order in which we add or multiply numbers does not change the result, so \(8x - 3x = 3x + 8x - 3x\). Then, the associative property allows us to group numbers differently without changing the results: \(8x - 3x = (8x - 3x)\). We don't need the distributive property for this problem, but it's good to note that it lets us multiply a number by a group of numbers added together (\(a(b + c) = ab + ac\)).

Utilizing these properties effectively is crucial for manipulating and simplifying equations to reach a solution.