Problem 2 - Introductory Algebra for College Students

Question

A number increased by 60 is equal to \(410 .\) Find the number.The sum of a number and 43 is \(107 .\) Find the number.

Step-by-Step Solution

Answer

The unknown number in the first problem is 350, and the unknown number in the second problem is 64.

Step 1: Problem 1: Equation setup

First, translate the word problem into an algebraic equation. The problem states 'A number increased by 60 is equal to 410', so the equation becomes: x+60=410.

Step 2: Problem 1: Solve the equation

To find the unknown number x, simplify the equation by subtracting 60 from both sides: x=410-60; x=350.

Step 3: Problem 2: Equation setup

Translate the second word problem into the algebraic equation, this problem is stating 'The sum of a number and 43 is 107', this equation becomes: x+43=107.

Step 4: Problem 2: Solve the equation

Like the first problem, simplify the equation by subtracting 43 from both sides: x=107-43; x=64.

Key Concepts

Translating Word Problems into Equations

To successfully solve algebra word problems, it's crucial to convert the words into mathematical statements, known as equations. This process begins by identifying what we're looking for, typically represented by a variable, like 'x'.

In our exercise, phrases such as 'a number increased by 60' and 'the sum of a number and 43' tell us we're adding to our unknown number, which we've named 'x'. This leads to the creation of algebraic expressions—'x + 60' and 'x + 43'. Each expression is then set equal to the given totals, resulting in the equations 'x + 60 = 410' and 'x + 43 = 107'.

To become proficient at this conversion, practice by breaking down the words:

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operation symbols. It does not have an equality sign, which differentiates it from an equation.

In the context of our exercise, 'x + 60' and 'x + 43' are algebraic expressions where 'x' is the variable representing the unknown number we're trying to find, and '60' and '43' are known numbers added to our variable. Understanding how to form and manipulate these expressions is key to solving algebra problems. Here's a bullet-point list to help you remember what goes into an algebraic expression:

• Variables: represent unknowns.
• Constants: known values that stay the same.
• Operators: symbols like '+', '-', '*', '/', which determine the operations to be performed on the variables and constants.

By mastering the art of combining these elements, you create a toolset to tackle a variety of problems.

Equation Simplification

After setting up the appropriate equations, simplification is the next critical step to find the solution. Simplifying an equation means performing operations to isolate the variable and uncover its value.

Examples from our exercise include subtracting '60' from 'x + 60 = 410' and '43' from 'x + 43 = 107'. Both actions simplify the equations to 'x = 350' and 'x = 64', respectively, making the solutions clear.

Essentially, simplification involves:

• Balancing equations by performing the same operation on both sides.
• Combining like terms, if any are present.
• Reducing expressions to their simplest form to reveal the solution.

Through consistent practice of these techniques, equation simplification becomes a straightforward process enabling you to quickly find accurate solutions to algebra problems.