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Isolating variables is a crucial step in solving algebraic equations. This technique ensures that the variable we are solving for is alone on one side of the equation.
This means we need to move all other terms and coefficients to the opposite side.
In the given problem, we need to isolate the variable \(y\). In the equation \(9x + y = 13\), the term \(9x\) is on the same side as \(y\).
To isolate \(y\), subtract \(9x\) from both sides:
$$y = 13 - 9x$$
By performing this subtraction, we have successfully isolated \(y\).
Now, \(y\) is alone on one side of the equation, which makes it easier to solve for its value.

Linear equations form the basis of many algebraic problems. A linear equation is any equation that can be written in the form \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constants.
In the given problem, \(9x + y = 13\) is a linear equation.
Linear equations represent straight lines when graphed on a coordinate plane. The general form for linear equations involving two variables is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
In our solution, after isolating \(y\), the equation takes the form:
$$y = -9x + 13$$
This is now in the slope-intercept form \(y = mx + b\), with \(m = -9\) and \(b = 13\). Here, -9 is the slope, and 13 is the y-intercept, meaning the line crosses the y-axis at 13.

Algebraic manipulation involves rearranging and simplifying equations to solve for the unknown variables.
It includes various techniques like adding, subtracting, multiplying, or dividing both sides of an equation by the same value.
In the given problem, we used subtraction to isolate \(y\) by removing the \(9x\) term:
$$9x + y - 9x = 13 - 9x$$
which simplifies to: $$y = 13 - 9x$$
This step-by-step manipulation makes it easier to understand how each part of the equation relates to the final solution.
Mastering algebraic manipulation is essential for solving more complex equations and for understanding higher-level math concepts.