A system of equations has infinite solutions when the two equations are equivalent, meaning they represent the same line. Graphically, this would mean that the two lines overlap each other perfectly, showing that they are not two separate lines, but the same line.
To show that the system of equations y = 3x + 1 has infinite solutions, plug in the points (3,10) and (0,1) into the equation. For both points, y = 3(3) + 1 = 10 and y = 3(0) + 1 = 1. This shows that both points lie on the line y = 3x + 1, indicating that there are infinitely many solutions.
A system of equations has no solution when the two lines are parallel and do not intersect. Graphically, this would mean that the lines do not meet at any point.
To show that the system of equations y = 5x + 2 and y = -4x - 1 has no solution, plug in the points (1,1) and (0,2) into both equations. For the first equation, y = 5(1) + 2 = 7, but for the second equation, y = -4(1) - 1 = -5. These points do not lie on the same line, indicating that there are no solutions.
A system of equations has one solution when the two lines intersect at a single point. Graphically, this means that the two lines cross each other at a single point.
For the system of equations y = 5x - 4 and y = -2x + 3, find the intersection point by setting the two equations equal to each other: 5x - 4 = -2x + 3. Solving for x gives x = 1. Substituting x = 1 back into one of the equations (y = 5(1) - 4 = 1), shows that the solution is (1,1). This confirms there is one unique solution. Graphically, if the two lines were plotted, they would intersect at (1,1), demonstrating that there is one solution.
1. Y=5x-4 (one solution)
2. Y=-2x+3 (one solution)
3. Y=3x+1 (infinite solutions)
4. Y=3x+1 (infinite solutions)
5. Y=5x+2 (no solutions)
6. Y=-4x-1 (no solutions)
What does it mean for a system to have infinite solutions? Explain what it means graphically to have infinite solutions.
Plug in (3,10) and (0,1) and check to show that the system of equations has more than on solution. Explain your work.
What does it mean for a system to have no solution? Explain what it means graphically for a system to have no solution.
Plug in (1,1) and (0,2) and check to show that the system of equations has no solution. Show your work.
What does it mean for a system to have one solution? Explain what it means to graphically for a system to have one solution.
Identify the solution. Plug the solution into the equation and check to show that the system has one solution. Explain how if the lines were graphed would show the system has one solution.
1 answer