Problem 66 - Elementary Algebra

Question

Solve the equation \(4 y+5=-3\).

Step-by-Step Solution

Answer

Answer: The solution is \(y = -2\).

Step 1: Move the constant to the right side

Subtract 5 from both sides of the equation: \(4y + 5 - 5 = -3 - 5\) This simplifies to: \(4y = -8\)

Step 2: Divide by the coefficient of y

Divide both sides of the equation by 4: \(\frac{4y}{4} = \frac{-8}{4}\) This simplifies to: \(y = -2\) The solution of the equation \(4y + 5 = -3\) is \(y = -2\).

Key Concepts

Linear Equations

Linear equations are like math problems that involve finding the value of a variable, typically shown as \( x \), \( y \), or another letter that makes the equation true. These are called "linear" because if you were to graph them, they would produce a straight line. In the case of the equation \( 4y + 5 = -3 \), it's a straightforward linear equation with one variable, which is \( y \).

Linear equations can be written in the standard form \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants (numbers), and \( x \) is the variable. The goal is to find which value of \( x \) makes the equation true. Linear equations can be simple like this one or part of more complex systems.

Solving Equations

When we talk about solving equations, we refer to the process of finding the value of the variable that makes the equation true. It's like a puzzle you solve by finding pieces that fit.

To solve linear equations, we often follow a series of logical steps. For example, with the equation \( 4y + 5 = -3 \):

• First, "move" numbers around to isolate the variable. This means you get \( y \) by itself on one side of the equation.
• Equations should always remain balanced, so whatever operation you do on one side, you must do on the other as well.
• Operations include adding, subtracting, multiplying, or dividing both sides of the equation by the same number.

By rearranging and simplifying the work, you'll get to a point where the variable is isolated, like how we reached \( y = -2 \) in the above example.

Algebraic Manipulation

Algebraic manipulation is the ability to use various operations and mathematical rules to transform equations or expressions into a more manageable or simplified form. This skill is hugely beneficial in solving equations. It's like using tools to rearrange parts of a math problem until you solve it.

Let's break down the example \( 4y + 5 = -3 \):

• Start by removing any constants from the side of the equation with the variable. In this example, subtracting 5 from both sides helps simplify the equation to \( 4y = -8 \).
• Next, handle the coefficient of the variable. This involves dividing or multiplying both sides to isolate the variable. Here, we divided both sides by 4 to solve for \( y \), resulting in \( y = -2 \).

Through these steps, you manipulate the equation into a form where you can easily find the solution, which develops your mathematical intuition.