graphing and substitution systems

Front

What is the substitution method in solving systems of equations?

Back

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

Front

How do you solve the equation -4x + 4y = 8 for y?

Back

To solve for y, rearrange the equation: 4y = 4x + 8, then divide by 4: y = x + 2.

Front

What is the solution to the system of equations y = 2x + 1 and y = 4x - 1?

Back

The solution is (1, 3), found by setting the two equations equal to each other and solving for x.

Front

What does it mean if a system of equations has no solution?

Back

A system has no solution if the lines represented by the equations are parallel and never intersect.

Front

How can you determine if a system of equations has one solution, no solution, or infinitely many solutions?

Back

By analyzing the slopes and y-intercepts of the lines: if they have the same slope and different intercepts, there is no solution; if they have the same slope and the same intercept, there are infinitely many solutions; if they have different slopes, there is one solution.

Front

What is the first step in using substitution to solve a system of equations?

Back

The first step is to solve one of the equations for one variable in terms of the other variable.

Front

How do you graph the solution of a system of equations?

Back

To graph the solution, plot the lines represented by the equations on the same coordinate plane and identify the point(s) where they intersect.

Front

What is the significance of the point (1, -4) in the context of a system of equations?

Back

The point (1, -4) is the solution to the system of equations, meaning it satisfies both equations.

Front

What is the equation of a line in slope-intercept form?

Back

The slope-intercept form of a line is given by y = mx + b, where m is the slope and b is the y-intercept.

Front

How do you check if a solution (x, y) is correct for a system of equations?

Back

To check if (x, y) is a solution, substitute x and y into both equations and verify that both equations hold true.