• Definition: A linear equation represents a straight line when graphed on a coordinate plane in which the highest power of any variable is 1. Every term will be either a constant or a factor of a constant with a first order term.
• General Expression: $y = mx+b$, where
  • $m$ and $b$ are constants.
  • $x$ represents the variable

Intercept is a point where a line crosses an axis.

• Set $y = 0$ to find x-intercept.
  • The x-intercept of a line is the point where the line crosses the x-axis. This means that the y-coordinate of the x-intercept is always 0.
• Set $x = 0$ to find the y-intercept.
  • The y-intercept of a line is the point where the line crosses the y-axis. This means that the x-coordinate of the y-intercept is always 0.

• Definition: The rate of change (steepness) between Y and X.
• Calculation: It is calculated by dividing the change in the y-coordinate of two points on the line by the change in the x-coordinate of the same two points.
• Formula: $\frac{\Delta Y}{\Delta X}, \frac{Y_2-Y_1}{X_2-X_1}$

• Positive Slope: For every increase in $x$, $y$ increases.
• Negative Slope: For every increase in $x$, $y$ decreases.

The phrase means describing how the quantity $y$ changes for a corresponding change in $x$

• A horizontal slope occurs when there is no variation in Y as X increases, indicating a slope of 0.
• A vertical slope occurs when there is no variation in X as Y increases, resulting in an undefined slope.

$ax+by=c$, where..

• $a,b$ are constants, real numbers and not equal to 0.
• $x$ and $y$ are the variables.

There are always infinitely many solutions to a linear equation in two variables.

To identify the two points, solve the equation by substituting 0 for $x$. For the next point, substitute 0 for $y$ again.

$ax = r, x = \frac{r}{a}.$ The set of the solution is restricted the $x$ variable to be equal to $\frac{r}{a}.$, which is a constant.

A system of linear equations is a collection of two or more linear equations.

• A system of linear equations with one solution has a unique intersection point.
• A system of linear equations with no solutions is inconsistent.
• A system of linear equations with infinitely many solutions has overlapping lines.

Eliminate one variable to solve to the other.

1. Multiply or divide one or both equations to make one of the variables equal in both equations.
2. Add or subtract to eliminate one of the variables.
3. Solve the remaining variable.
4. Substitute the solved variable into one of the original equation.

Substitute the other constraint (equation) into the other equation vice-versa.

1. Solve one equation for one variable.
2. Substitute one expression into the other equation and simplify the equation.
3. Substitute the solved variable into one of the original equation to find the other variable.

• Independent equations means the equations don’t affect each other’s solution
• Dependent equations have solutions that are connected or influenced by each other.

• Consistent equations have at least one solution. Whereas, inconsistent equations do not have a solution.

• A system of equations is dependent if it has infinitely many solutions, all of which form a linear relationship with each other.
  • Both equation have the same slope and same $y$ and $x$ intersection.
• A system of equations is independent if it has a unique solution, meaning there is only one set of values for the variables that satisfies all equations in the system.
  • Both equation have different slope but the lines will intersect at one $(x,y)$ value.

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