/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 103 Solve each equation for \(y\) ... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 103

Solve each equation for \(y\) $$ 6 x+y=4 $$

Short Answer

Expert verified
y = 4 - 6x

Step by step solution

01

Isolate the variable y

Start by isolating the variable y on one side of the equation. To do this, subtract 6x from both sides of the equation.
02

Perform the subtraction

Subtract 6x from 4 on the right side of the equation to get: \( y = 4 - 6x \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolating Variables
Understanding how to isolate variables is key when solving linear equations. Isolating the variable means rewriting the equation so that the variable is alone on one side.
Here's how you can isolate a variable:
  • Identify the variable you need to isolate. In this exercise, it's \(y\).
  • Perform operations that help you move all other terms to the opposite side. You can add, subtract, multiply, or divide both sides of the equation.
  • Simplify the equation, ensuring the variable stands alone with a coefficient of 1 if possible.

For the given equation, \[ 6x + y = 4 \], we isolate \(y\) by moving \(6x\) to the other side. We subtract \(6x\) from both sides, resulting in \[ y = 4 - 6x \]. Now, \(y\) is isolated.
Linear Equations
Linear equations are equations of the first order. They make a straight line when graphed. The general form is \[Ax + By = C \], where \(A\), \(B\), and \(C\) are constants.
Several properties of linear equations include:
  • The solution to a linear equation is any pair of \(x\) and \(y\) values that make the equation true.
  • There is exactly one solution (a single point on the graph), infinitely many solutions (a line), or no solutions (parallel lines).
  • The equation represents a balance, meaning whatever you do to one side, you must do to the other.

In this exercise, the linear equation given is \[6x + y = 4 \], a clear example of a first-order equation that can be graphed as a line once solved.
Basic Algebra
Basic algebra involves arithmetic with symbols (like x, y) instead of numbers. It's foundational for higher-level math concepts. Here are some core principles:
  • Variables: Symbols representing numbers. For example, \(x\) and \(y\) could be any number.
  • Constants: Numbers on their own. In this exercise, \(4\) is a constant.
  • Operations: Addition, subtraction, multiplication, and division are used to manipulate equations.

When we solve equations, we apply these operations to simplify expressions and find the values of variables.
In the step-by-step exercise, subtracting \(6x\) from both sides is an application of basic algebra to isolate the variable \(y\). This skill of manipulating equations is crucial for solving all kinds of algebraic problems.

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