Solve Systems using Substitution

Front

What is a solution to a system of equations?

Back

It's the point that solves both equations at the same time.

Front

How do you solve the system of equations y=8x+1 and y=6x+3 using substitution?

Back

Substitute y from one equation into the other to find x, then back-substitute to find y. The solution is (1, 9).

Front

What is the first step when solving the system 3x + 3y = 3 and y = 3x + 5?

Back

The variable is already isolated for y, so substitute it into the first equation: 3x + 3(3x + 5) = 3.

Front

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

Back

It means the lines represented by the equations are parallel and never intersect.

Front

How do you solve the system x = 6 and y = 2x - 3 using substitution?

Back

Substitute x = 6 into the second equation: y = 2(6) - 3, which gives y = 9. The solution is (6, 9).

Front

What is substitution in the context of solving systems of equations?

Back

Substitution is a method where you solve one equation for one variable and then substitute that expression into another equation.

Front

What is the graphical representation of a system of equations?

Back

The graphical representation consists of lines on a coordinate plane, where the intersection point(s) represent the solution(s).

Front

What does it mean for two equations to be dependent?

Back

Dependent equations have infinitely many solutions, as they represent the same line.

Front

What is the difference between independent and dependent systems of equations?

Back

Independent systems have exactly one solution (the lines intersect at one point), while dependent systems have infinitely many solutions (the lines are the same).

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 slopes are different, one solution; if slopes are the same and intercepts differ, no solution; if slopes and intercepts are the same, infinitely many solutions.