Solutions to Linear Equations

Front

What is a linear equation?

Back

A linear equation is an equation that represents a straight line when graphed. It can be written in the form y = mx + b, where m is the slope and b is the y-intercept.

Front

What does the slope of a line represent?

Back

The slope of a line represents the rate of change of the y-coordinate with respect to the x-coordinate. It indicates how steep the line is and the direction it goes (positive or negative).

Front

What is the significance of two linear equations having different slopes?

Back

If two linear equations have different slopes, they will intersect at exactly one point, meaning they have one unique solution.

Front

What happens when two linear equations have the same slope but different y-intercepts?

Back

When two linear equations have the same slope but different y-intercepts, they are parallel lines and will never intersect, meaning they have no solutions.

Front

What is the graphical representation of a system of linear equations?

Back

The graphical representation of a system of linear equations consists of the lines corresponding to each equation plotted on a coordinate plane.

Front

How can you determine the number of solutions in a system of linear equations?

Back

The number of solutions can be determined by analyzing the slopes and y-intercepts of the equations: one solution (intersecting lines), no solution (parallel lines), or infinitely many solutions (same line).

Front

What is the y-intercept in a linear equation?

Back

The y-intercept is the point where the line crosses the y-axis, represented by the value of b in the equation y = mx + b.

Front

What does it mean for a system of equations to be consistent?

Back

A system of equations is consistent if it has at least one solution, which can be either one unique solution or infinitely many solutions.

Front

What does it mean for a system of equations to be inconsistent?

Back

A system of equations is inconsistent if it has no solutions, typically represented by parallel lines.

Front

What is the method of substitution in solving linear equations?

Back

The method of substitution involves solving one equation for one variable and then substituting that expression into the other equation to find the values of both variables.