Problem 69 - Algebra 2

Question

OPEN ENDED Write a two-step equation with a solution of \(-7 .\)

Step-by-Step Solution

Answer

Create 2x + 3 = -11; solving gives x = -7.

Step 1: Set up Equation Structure

To create a two-step equation, we need an equation that involves a variable and requires two operations to isolate that variable. We'll use the variable \(x\). Begin with a simple equation: \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants.

Step 2: Choose Constants for the Equation

Since we know the solution is \(x = -7\), choose \(a = 2\) and \(b = 3\) which leads us to: \(2x + 3 = c\). Plug \(x = -7\) into this equation to determine \(c\).

Step 3: Solve for \(c\) Using the Solution

Substitute \(x = -7\) into the equation \(2x + 3 = c\). Calculate: \(2(-7) + 3 = c\). Simplify to get \(-14 + 3 = -11\). Thus, \(c = -11\).

Step 4: Write the Final Equation

The final two-step equation is \(2x + 3 = -11\). Solving it should yield \(x = -7\), verifying its correctness.

Key Concepts

Equations with variables

An equation with a variable is like a puzzle. Variables are symbols that stand for unknown numbers, and often letters like \( x \) or \( y \) are used. When we talk about equations with variables, we analyze how these variables relate to specific numbers on either side of the equation.

For instance, in our example equation \( 2x + 3 = -11 \), the term \( 2x \) represents a variable term. It comprises a variable \( x \) and a coefficient \( 2 \), showing that this term depends on the value of \( x \). This means that when \( x \) changes, the value of \( 2x \) changes too.

The purpose of an equation is to find the specific value that the variable represents, leading to a true statement. In simpler words, it's about solving the mystery with the clues given in the equation.

Solving equations

Solving equations is like being a detective to find the unknown value that makes an equation true. When we solve an equation, we perform specific operations to isolate the variable on one side of the equation. This is known as "solving for the variable."

Here are the basic steps to solving our two-step equation \( 2x + 3 = -11 \):

• First, deal with addition or subtraction to simplify the equation. Subtract \( 3 \) from both sides to get \( 2x = -14 \).
• Second, handle multiplication or division to completely isolate the variable. Divide both sides by \( 2 \) to get \( x = -7 \).

This step-by-step approach helps us focus on reversing operations, systematically working our way to solving the equation altogether. It's important to do the same operation on both sides to maintain balance in the equation.

Linear equations

Linear equations are the foundation of algebra. They describe a straight line when visualized on a graph. These equations take the form \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable. They are called "linear" because they form a line with a constant slope when plotted.

Our example \( 2x + 3 = -11 \) is a linear equation. The coefficient \( 2 \) before \( x \) tells us about the slope, while the constant \( 3 \) acts as the y-intercept when drawn on a Cartesian plane. When we solve this equation, we're essentially finding where this line would intersect with the horizontal axis, indicating the value of \( x \) that satisfies the equation, which in this case is \( -7 \).

Understanding linear equations not only helps in solving individual problems, but also forms the building blocks for more complex algebraic concepts. It's all about finding harmony within the equation's structure.