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When solving linear equations, one of the first steps is often to isolate the variable you want to solve for. In this case, we are isolating the variable 'y'.
Isolating a variable means rearranging the equation so that 'y' is alone on one side of the equal sign.
Let's break it down further: Start by moving all other terms (those not containing 'y') to the other side of the equation. This is usually done by performing the same operations on both sides of the equation to keep it balanced.
In our example equation, we have: \(6x - 5y = 7\). To isolate the term with 'y', we subtract \(6x\) from both sides: \[ \begin{align*} 6x - 5y - 6x &= 7 - 6x \end{align*} \]
This simplifies to: \( -5y = 7 - 6x \). Now, the term with 'y' is isolated from the other terms in the original equation.

After isolating the variable 'y', the next step is to solve for 'y'. This means we want 'y' by itself, with no coefficients (numbers multiplied by 'y'). To do this, you'll need to perform operations that will get rid of any coefficient in front of 'y'.
In the simplified equation \( -5y = 7 - 6x \), notice that 'y' is multiplied by -5. To solve for 'y', divide both sides of the equation by -5: \[ \begin{align*} y &= \frac{7 - 6x}{-5} \end{align*} \]
By dividing each side by -5, we effectively cancel out the -5 that was multiplying 'y'. Now, to simplify it further, we distribute the negative sign in the denominator across the numerator: \[ \begin{align*} y &= \frac{-7 + 6x}{5} \end{align*} \]
Or equivalently: \[ \begin{align*} y &= \frac{6x - 7}{5} \end{align*} \]
This gives us the final expression for 'y' in terms of 'x'.

A linear equation is an equation of the first degree, meaning it has no exponents higher than one. It can be written in the general form: \( Ax + By = C \), where 'A', 'B', and 'C' are constants, and 'x' and 'y' are variables.
The term 'linear' refers to the fact that the graph of such equations is a straight line. Consider our equation \(6x - 5y = 7\). This is a typical linear equation.
Working with linear equations generally involves:

• Isolating the variable you want to solve for
• Performing arithmetic operations to simplify
• Rearranging the equation into a solvable format

Linear equations are fundamental in algebra because they appear in various real-world scenarios, from calculating distances to budgeting finances.
Understanding linear equations requires mastering the basics: knowing how to manipulate equations, isolate variables, and solve for them.