Solving Linear Equations and Inequalities

Front

What is a linear equation?

Back

A linear equation is an equation of the first degree, meaning it has no exponents greater than one. It can be written in the form ax + b = c, where a, b, and c are constants.

Front

What does it mean for a linear equation to have one solution?

Back

A linear equation has one solution when the graph of the equation intersects the x-axis at exactly one point.

Front

What is an example of a linear equation with one solution?

Back

12x = 0 has one solution: x = 0.

Front

What does it mean for a linear equation to have no solution?

Back

A linear equation has no solution when the lines represented by the equations are parallel and never intersect.

Front

What is an example of a linear equation with no solution?

Back

The equation 6x + 4 = 6x - 4 has no solution because it simplifies to 4 = -4, which is false.

Front

What does it mean for a linear equation to have infinite solutions?

Back

A linear equation has infinite solutions when the two sides of the equation are equivalent for all values of the variable.

Front

What is an example of a linear equation with infinite solutions?

Back

The equation 2(4x + 8) = 8x + 10 simplifies to 8x + 16 = 8x + 10, which is false, indicating infinite solutions.

Front

How do you solve the equation 10x + 2x + 5 = 8x + 4x + 7?

Back

Combine like terms: 12x + 5 = 12x + 7. This simplifies to 5 = 7, which is false, indicating no solution.

Front

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

Back

The coefficient represents the rate of change of the variable in the equation. It affects the slope of the line when graphed.

Front

What is the solution to the equation 10x - 3 = 3x + 32?

Back

To solve, combine like terms: 10x - 3x = 32 + 3, which simplifies to 7x = 35. Thus, x = 5.