Understanding equation of a straight line y = mx + c

A straight line is fixed by two things: how tilted it is, and where it sits on the grid. The form $y = mx + c$ captures both at once, which is why almost every question about lines starts by getting the equation into this shape.

The slope $m$ is the number multiplying $x$. It measures how much $y$ changes for every increase of $1$ in $x$: a positive $m$ rises from left to right, while a negative $m$ falls. The larger the size of $m$, the steeper the line.

The y-intercept $c$ is the constant term. It is the point where the line crosses the y-axis, which happens when $x = 0$. Substituting $x = 0$ into $y = mx + c$ leaves $y = c$.

Most equations you meet are not already in this form, for example $2x + 3y = 6$ or $y - 4 = 2(x - 3)$. The single skill that unlocks them all is rearranging to isolate $y$: move the $x$-term and any constants to the other side, then divide every term by whatever multiplies $y$. As soon as the equation reads $y = mx + c$, the slope and intercept are simply the two numbers in front of $x$ and standing alone.

This one form also powers later ideas. Parallel lines share the same slope $m$, so comparing slopes tells you whether two lines are parallel. And to check whether a point lies on a line, substitute the point into the equation and see whether both sides come out equal.

Write $4x + 2y = 10$ in the form $y = mx + c$.
Subtract $4x$ from both sides to get $2y = -4x + 10$, then divide every term by $2$: $y = -2x + 5$.
Comparing with $y = mx + c$, the slope is $m = -2$ and the y-intercept is $c = 5$.

Given $y - 5 = 3(x - 1)$, find the slope and y-intercept.
Expand and isolate $y$: $y = 3x - 3 + 5$, which gives $y = 3x + 2$.
Comparing with $y = mx + c$, the slope is $m = 3$ and the y-intercept is $c = 2$.