Problem 5 - Algebra Form and Function

Question

Solve the equations. $$ 4 p-1.3=-6 p-16.7 $$

Step-by-Step Solution

Answer

Answer: p = -1.54

Step 1: Combine like terms

Combine the terms with p on both sides of the equation: $$ 4p + 6p = -16.7 + 1.3 $$

Step 2: Simplify the equation

Simplify both sides of the equation: $$ 10p = -15.4 $$

Step 3: Solve for the variable

Divide both sides of the equation by the coefficient of the variable (10) to solve for p: $$ p = \frac{-15.4}{10} $$

Step 4: Simplify the answer

Simplify the value of p to get the final answer: $$ p = -1.54 $$ Thus, the solution to the equation is \(p = -1.54\).

Key Concepts

Algebra

Algebra is essentially the branch of mathematics that uses symbols and letters to represent numbers and quantities in equations and formulas. This allows us to solve problems where some values are unknown. In algebra, we work with variables, which are placeholders for these unknown values. Understanding algebra lays the foundation for more complex math topics, and it's used in everyday problem solving. Key concepts of algebra include:

• Variables: Symbols like \(p\) in our equation, that stand in for unknown numbers.
• Expressions: Combinations of variables and numbers, such as \(4p - 1.3\).
• Equations: Statements that show the equality of two expressions, which we solve to find the value of a variable.

In our exercise, algebra helps us establish relationships between these expressions to ultimately solve for the variable \(p\). With practice, these concepts become intuitive, making it easier to tackle various math challenges.

Linear Equations

Linear equations are a major component of algebra, involving equations of the first order. This means the variable is not raised to any power higher than one. A linear equation looks like \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is a variable. In linear equations:

• They graph as straight lines on a coordinate plane.
• They have one solution because we're dealing with a single variable.
• They require combining like terms and isolating the variable.

For our specific problem, \(4p - 1.3 = -6p - 16.7\), we combine like terms to formulate a single linear equation, making it easier to solve. This method of consolidation is crucial in managing the equation's complexity and simplifying it for easy interpretation.

Variable Manipulation

Variable manipulation involves handling variables to simplify equations and ultimately find their solutions. This process includes rearranging equations, using operations like addition, subtraction, multiplication, and division to isolate the variable.Steps of variable manipulation involve:

• Moving terms with variables to one side of the equation: For example, adding \(6p\) to both sides in our exercise.
• Simplifying the equation: Once like terms are combined, such as \(4p + 6p\), we simplify to \(10p\).
• Isolating the variable: Divide both sides by 10, the coefficient of \(p\), giving \(p = -1.54\).

By systematically breaking down the steps, variable manipulation makes even complicated equations manageable. The goal is always to have the variable on one side and its value on the other for clear understanding.