Check and Graph Solutions to Two-Variable Equations

what is the resulting combined equation for the system below?

• -18x = 18

• y = 18

• 2y= 18

• 18x + 2y = 18

Which method would be easiest to use for this problem?

• Graphing

• Elimination

• Substitution

Given the system of equations, what would I multiply the first equation by to eliminate the x variable?

• 4

• 2

• 1

• -2

When solving the following systems of equations by substitution, which of the following would be the correct first step?

A system of equations consists of two of more linear equations. In Algebra I, we only explore systems with two equations. The solution of a system of equations is the solution that both equations share in common. Each linear equation has a set of solutions. For example, each point on a line is a solution for the equation of that line. If two lines intersect, the point of intersection is the solution that the two equations share in common. Thus, the intersecting point is the solution to the system of equations. The solution of a system of equations is _____.

• The y-intercept

• The point where the two lines intersect

• The numerical value of the slope

• The difference of the two slopes

What is the solution to this system of equations?

• (6, 1)

• (-1, 6)

• (6, -1)

• (1, 6)

• 
• 
• 
• 

When the line graphs of a linear system intersect, that point is the solution of the system. However, sometimes the line graphs of a system do not intersect. There are two situations when this is true. First, if the lines are parallel, they will not intersect. In this situation, there is no solution. Second, one line lies on top of the other line. Therefore, the lines are always touching. In this situation, there are infinitely many solutions. Check all true answers.

• A system of equations, whose graphs are parallel lines, has no solution.

• A system of equations, whose graphs appears to be one line, has infinitely many solutions.

• A system of equations, whose graphs intersect, has one and only one solution.

• A system of equations will have one of three possible solutions (1) exactly one, (2) no solution, or (3) infinitely many solutions.

A system of equations may be solved by graphing or algebraically. If solved by graphing, observe the graph to determine the solution: (1) if the lines intersect, the point of intersection is the solution, (2) if the lines are parallel, there is no solution, and (3) if there appears to be only one line, there are infinitely many solutions. If you solve algebraically, the values of x and y will determine the solution: (1) if a value is found for x and y, those values are written as an ordered pair, which is the solution, (2) if the x and y values are eliminated and the result is a false statement (such as 3 = 0), there is no solution, and (3) if the x and y values are eliminated and the result is a true statement (such as 3 = 3), there are infinitely many solutions. Select all true statements pertaining to solving a system algebraically.

• When solving, if the final answer produces a value for x and y, this system has one solution.

• When solving, if the final answer produces a false statement, this system has no solution.

• When solving, if the final answer produces a true statement, this system has infinitely many solutions.

• When solving, if the final answer produces a fraction, you have made a mistake.

• (6, 0)

• (4, 6)

• (6, 4)

• (0, 6)