Problem 26 - Algebra 1

Question

Solve the equation. $$-7 x+4 x=9$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = -3\).

Step 1: Simplify the equation

Start by simplifying the equation. Simplify the left-hand side by combining the terms: \(-7x\) and \(4x\) give: \(-3x\), so the simplified equation is now \(-3x = 9\).

Step 2: Divide by -3

To solve for \(x\), divide both sides by \(-3\). This gives the equation \(x = -3 \).

Key Concepts

Simplifying Equations

Simplifying equations is a crucial step to solve any algebraic equation effectively. It involves making the equation less complicated and easier to work with. You kick-start this process by looking at each side of the equation and removing any unnecessary components. One way of simplifying equations is by combining like terms, which will be discussed further on.

• Ensure that any parentheses are eliminated by distributing multiplication if needed.
• Work on reducing fractions or coefficients where possible to keep numbers easy to manage.

When this process is done correctly, your equation will be much more straightforward and manageable, forming a clean basis for further operations.

Combining Like Terms

Combining like terms is an essential skill when solving equations. Like terms are terms that have the same variable raised to the same power. For instance, in the equation \(-7x + 4x = 9\), both \(-7x\) and \(4x\) are like terms because they share the variable \(x\).

To combine these, you need to add their coefficients: \(-7 + 4 = -3\). This reduces the equation to \(-3x = 9\).

• Keep an eye out for terms that have the same variable.
• Make sure to apply the correct arithmetic operation to their coefficients.
• Remember that constants (numbers without variables) can also be combined with other constants.

By effectively combining like terms, you simplify the equation, making the path to the solution much clearer.

Isolating Variables

Isolating the variable is a method used to get the unknown variable by itself on one side of the equation. The purpose of this method is to solve the equation for that variable. After combining like terms in our example equation \(-3x = 9\), the next step is to isolate \(x\). You achieve this by performing the same mathematical operation on both sides of the equation.

In this specific scenario, you need to divide both sides by \(-3\) to isolate \(x\): \(x = 9 \div -3 = -3\).

• Always perform the same operation on both sides of the equation to maintain balance.
• Use inverse operations to get the variable alone (for multiplication, divide and vice versa).
• Double-check your work by substituting your solution back into the original equation.

Once the variable is isolated, you will have your solution, completing the process of solving the equation.