linear equations

Answer 1

Linear equations are mathematical statements that describe a relationship between two variables in a way that produces a straight line when graphed. The general form of a linear equation in two variables (x and y) can be written as:

$$y=mx+b$$

Where:

• $y$ is the dependent variable.
• $x$ is the independent variable.
• $m$ is the slope of the line (the rate of change) that describes how much $y$ changes for a one-unit change in $x$.
• $b$ is the y-intercept, the value of $y$ when $x=0$.

Key Concepts:

1. Slope (m): Determines the steepness and direction of the line:

   • Positive slope ($m>0$): The line rises as it moves from left to right.
   • Negative slope ($m<0$): The line falls as it moves from left to right.
   • Zero slope ($m=0$): The line is horizontal.
   • Undefined slope: The line is vertical.

2. Y-intercept (b): The point at which the line crosses the y-axis.

3. Standard Form: A linear equation can also be expressed in standard form: $$Ax+By=C$$ where $A$, $B$, and $C$ are constants, and $A$ and $B$ are not both zero.

Examples:

1. Example 1:

   • Equation: $y=2x+3$
   • Slope $m=2$, y-intercept $b=3$

2. Example 2:

   • Equation: $4x-2y=8$
   • To express in slope-intercept form, solve for $y$:
   • $-2y=-4x+8$
   • $y=2x-4$ (Here, $m=2$ and $b=-4$)

Solving Linear Equations:

To solve linear equations, you can use various methods, including substitution, elimination, and graphing.

Example of Solving via Substitution:

• Given a system of equations:

  1. $y=2x+3$
  2. $x+y=9$

• Substitute the expression for $y$ from Equation 1 into Equation 2: $$x+(2x+3)=9$$ $$3x+3=9$$ $$3x=6\Rightarrow x=2$$

• Now substitute $x$ back into Equation 1: $$y=2(2)+3=7$$

• Solution: $x=2,y=7$

If you have more specific questions or examples you'd like to explore regarding linear equations, feel free to ask!