Problem 94 - Algebra and Trigonometry

Question

Solve for \(y: 7 x+3 y=18 .\) (Section \(1.3,\) Example 7 )

Step-by-Step Solution

Answer

The solution to the equation \(7x + 3y = 18\) for \(y\) is \(y = 6 - \frac{7x}{3}\)

Step 1: Isolate y Term

To start, aim to isolate the \(y\) term on one side of the equation. You can do this by subtracting \(7x\) from both sides of the equation. This gives you \(3y = 18 - 7x\).

Step 2: Solve for y

Next, divide the entire equation by 3 to solve for \(y\). This gives you \(y = \frac{18 - 7x}{3}\).

Step 3: Simplify the Equation

In step 3, distribute the denominator into the numerator to simplify. This simplifies to \(y = 6 - \frac{7x}{3}\)

Key Concepts

Isolate Variable

When solving linear equations, one of the primary goals is to isolate the variable of interest on one side of the equation to find its value. Isolating a variable means manipulating the equation so that the variable is by itself on one side, and everything else is on the other side.

In the provided exercise, the variable to isolate is 'y'. To initiate this, you would move all the terms containing 'x' to the opposite side of the equation, which in this case, is achieved by subtracting '7x' from both sides. It’s a strategic move that simplifies the equation, turning it into a more straightforward, solvable form where 'y' stands alone and is clearly the subject.

Algebraic Manipulation

Algebraic manipulation involves applying mathematical operations to an equation to alter its form without changing its meaning or solution. These operations can include adding, subtracting, multiplying, and dividing terms on both sides of an equation, as well as factoring and expanding expressions.

The equation in our exercise demonstrates such manipulations; after isolating 'y', dividing both sides of the equation by '3' balances the equation and further isolates 'y'. Here, dividing by '3' is an essential step in the process of solving for the variable. It’s not just about moving terms around, but also about understanding how these operations help to reveal the value of the variable.

Equation Simplification

Equation simplification is reducing an equation to its simplest form, making it easier to interpret and solve. The objective is to break down the equation into the most basic elements that still accurately represent the original equation.

In our example, the simplification process involves dividing the numeric part and the algebraic part of the numerator by the denominator separately. Thus, dividing '18' by '3' simplifies to '6', and '-7x' divided by '3' remains as an algebraic fraction. The simplified equation, 'y = 6 - \(\frac{7x}{3}\)', is more manageable and displays the direct relationship between 'x' and 'y'. Simplification can often reveal patterns or insights that might be hidden in a more complex representation.