Problem 57 - Algebra and Trigonometry

Question

Find all values of \(x\) such that \(y=0\) $$ y=4[x-(3-x)]-7(x+1) $$

Step-by-Step Solution

Answer

The value of \(x\) that makes the equation equal to zero is \(x = 19\).

Step 1: Simplify the Inner Brackets

Simplify the inner brackets of the equation, this gives: \(y = 4[x-(3-x)] - 7(x+1) = 4[x-3+x] - 7x - 7 = 4[2x - 3] - 7x - 7\)

Step 2: Distribute the Multiplication

Next step is to distribute the multiplication on the left side of the equation, it results into: \(y = 4*2x - 4*3 - 7x -7 = 8x - 12 - 7x - 7\)

Step 3: Simplify

The equation further simplifies to: \(y = 8x - 7x -12 -7 = x - 19 \)

Step 4: Set y to Zero

Set \(y = 0\) then solve the equation for \(x\), which provides: \(0 = x - 19 \Rightarrow x = 19\)

Key Concepts

Algebra

Algebra is a branch of mathematics that involves the use of symbols and letters to represent numbers and quantities in equations and formulas.

At its core, algebra enables us to solve for unknown values, often referred to as variables, like the variable x in the given problem. One of the primary objectives in algebra is to isolate the variable on one side of the equation to find its value. This process of solving an algebraic equation requires various steps including simplification and the application of algebraic properties such as the distributive property.

Understanding algebra is not just about memorizing steps; it involves recognizing patterns, applying logic, and making connections between different mathematical concepts. It lays the foundation for advanced mathematical studies and numerous applications in fields such as science, engineering, and economics.

Equation Simplification

The process of equation simplification is crucial in solving algebraic equations effectively. It involves reducing an equation to its simplest form, making it easier to solve.

Several strategies are used to simplify equations:

• Combining like terms, which are terms that contain the same variable raised to the same power.
• Eliminating parentheses through the distributive property or by simplifying the expression inside them first, as seen in the step-by-step solution provided.
• Moving terms from one side to the other, using the addition or subtraction of identical terms on both sides.

Simplification might seem straightforward, but it's essential to proceed methodically to avoid errors. By focusing on these strategies, students enhance their problem-solving skills and achieve a clearer path to solving for the unknown variable.

Distributive Property

The distributive property is a fundamental algebraic property that makes complex arithmetic much more manageable. It allows you to multiply a single term by each term within a set of parentheses, effectively distributing the multiplication over addition or subtraction.

Mathematically, the property is expressed as: \(a(b + c) = ab + ac\). In the context of the problem given, the distributive property is used in Step 2, where we multiply the number 4 outside the brackets by both terms inside the brackets \(2x\) and \(−3\), leading to \(8x - 12\).

Mastering the distributive property empowers students to handle complex equations more comfortably, breaking them down into simpler components that make finding the solutions less daunting.