Problem 37 - Algebra for College Students

Question

Solve each equation. \(5 x-4(x-6)=-11\)

Step-by-Step Solution

Answer

The solution is \(x = -35\).

Step 1: Distribute the Negative Sign

The equation is given as \(5x - 4(x - 6) = -11\). First, distribute the -4 through the parentheses: \(-4(x - 6) = -4x + 24\). Substitute it back into the equation, resulting in \(5x - 4x + 24 = -11\).

Step 2: Combine Like Terms

Simplify the equation by combining like terms. Combine \(5x\) and \(-4x\): \(5x - 4x = x\). This gives us: \(x + 24 = -11\).

Step 3: Isolate the Variable

To isolate \(x\), subtract 24 from both sides of the equation. It simplifies to: \(x = -11 - 24\).

Step 4: Simplify the Solution

Perform the arithmetic operation on the right side: \(-11 - 24 = -35\). Therefore, \(x = -35\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to simplify expressions and solve equations more easily. When you're faced with a term being multiplied by a group of terms inside parentheses, the distributive property helps to "distribute" the multiplication over each term inside the parentheses. For example, if you have the expression \(a(b + c)\), using the distributive property, you would multiply \(a\) by both \(b\) and \(c\), resulting in \(ab + ac\).

In the exercise \(5x - 4(x - 6) = -11\), the distributive property is applied to the term \(-4(x - 6)\).

• Distribute \(-4\) to both \(x\) and \(-6\).
• This results in the expression \(-4x + 24\).
• Replace the original term in the equation to continue solving: \(5x - 4x + 24 = -11\).

Mastering the distributive property will make it easier for you to simplify and solve complex equations.

Combining Like Terms

Combining like terms simplifies algebraic expressions, making it easier to work with them. Like terms refer to terms that have the same variable raised to the same power. Thus, they can be combined by adding or subtracting their coefficients, while keeping the variable part unchanged.

In the equation you worked on, \(5x - 4x + 24 = -11\), there are two like terms: \(5x\) and \(-4x\).

• Both terms have the variable \(x\).
• Add their coefficients: \(5 - 4\).
• This results in a simpler term: \(x\).

After combining the like terms, the equation becomes \(x + 24 = -11\), which is much easier to solve. This technique is essential for making sense of and simplifying algebraic expressions.

Isolating Variables

Isolating the variable is a crucial step in solving equations, especially linear equations. It involves manipulating the equation to get the variable you are solving for on one side of the equation, while all other terms are on the opposite side.

In the equation \(x + 24 = -11\), you need to isolate \(x\).

• Subtract 24 from both sides of the equation.
• This operation removes the constant from the left side: \(x = -11 - 24\).
• Isolation results in removing additional terms that may prevent easy identification of the solution.

By isolating the variable, you make the equation easier to solve, leading you directly to the value of the variable.

Solving Linear Equations

Linear equations, like the one you solved, represent a relationship between variables that forms a straight line when graphed. Solving these equations usually involves a few straightforward steps: distributing, combining like terms, and isolating variables.

For the equation given, the steps culminated in solving the equation by performing a final arithmetic operation, resulting in the solution \(x = -35\).

• After isolating the variable, perform the arithmetic operation on the constants: \(-11 - 24\).
• This calculation results in \(-35\), which is the solution for \(x\).
• Having the variable isolated first ensures clarity during this final step.

Understanding each of these steps in solving linear equations equips you to tackle more complex scenarios that involve similar steps.