Solving Equations with Special Cases

Front

What is the general form of a linear equation in one variable?

Back

The general form of a linear equation in one variable is ax + b = 0, where a and b are constants and a ≠ 0.

Front

How do you determine if a linear equation has one solution, no solution, or infinitely many solutions?

Back

A linear equation has one solution if it can be simplified to a single value for the variable. It has no solution if it leads to a contradiction (e.g., 0 = 5). It has infinitely many solutions if both sides of the equation are equivalent after simplification.

Front

What does it mean for an equation to have no solution?

Back

An equation has no solution when the two sides cannot be made equal, often resulting in a contradiction.

Front

What is the significance of the coefficient of the variable in a linear equation?

Back

The coefficient of the variable indicates the rate of change of the dependent variable with respect to the independent variable.

Front

What is the process for solving a linear equation?

Back

To solve a linear equation, isolate the variable on one side of the equation using inverse operations (addition, subtraction, multiplication, division) until the variable is alone.

Front

What is an example of an equation that has infinitely many solutions?

Back

An example is 2x + 3 = 2x + 3, which simplifies to 0 = 0, indicating that any value of x is a solution.

Front

What is the first step in solving the equation 10x - 9 = 2x + 63?

Back

The first step is to isolate the variable x by moving all terms involving x to one side and constant terms to the other side.

Front

What does it mean to 'isolate the variable' in an equation?

Back

Isolating the variable means rearranging the equation so that the variable is on one side by itself, making it easier to solve for its value.

Front

How can you check if your solution to an equation is correct?

Back

You can check your solution by substituting the value back into the original equation to see if both sides are equal.

Front

What is the solution to the equation 11x - 5 = 8x + 31?

Back

The solution is x = 12.