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Questions and Answers
If 4x - y = 5 is one of the equations, which could be the other equation?
If 4x - y = 5 is one of the equations, which could be the other equation?
- y = x - 7
- y = 3x + 2
- y = -4x + 5 (correct)
- y = 2x - 1
How many solutions does the linear system have for the equations y = 2x - 5 and -8x - 4y = -20?
How many solutions does the linear system have for the equations y = 2x - 5 and -8x - 4y = -20?
one solution (2.5, 0)
What is the solution for the system of equations y = -1/2x + 4 and 2y = -8?
What is the solution for the system of equations y = -1/2x + 4 and 2y = -8?
No solution
Muriel says she has written a system of two linear equations that has an infinite number of solutions. One of the equations is 3y = 2x - 9. Which could be the other equation?
Muriel says she has written a system of two linear equations that has an infinite number of solutions. One of the equations is 3y = 2x - 9. Which could be the other equation?
What is the solution to the system of equations 2x - y = 7 and y = 2x + 3?
What is the solution to the system of equations 2x - y = 7 and y = 2x + 3?
Which values of m and b will create a system of equations with no solution?
Which values of m and b will create a system of equations with no solution?
What value of b will cause the system y = 6x + b and -3x + 1/2y = -3 to have an infinite number of solutions?
What value of b will cause the system y = 6x + b and -3x + 1/2y = -3 to have an infinite number of solutions?
What is the best approximation for the solution to the system y = -3 and y = x - 0.8?
What is the best approximation for the solution to the system y = -3 and y = x - 0.8?
How many solutions does the linear system y = -6x + 2 and -12x - 2y = -4 have?
How many solutions does the linear system y = -6x + 2 and -12x - 2y = -4 have?
Which equation could be Henry's if Fiona wrote the linear equation y = 2/5x - 5 and they discovered that his equation had all the same solutions?
Which equation could be Henry's if Fiona wrote the linear equation y = 2/5x - 5 and they discovered that his equation had all the same solutions?
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Study Notes
Systems of Linear Equations: Key Concepts
- A system of equations with one solution includes lines that intersect at a single point.
- Example equation indicating one solution: 4x - y = 5, with an alternative solution presented as y = -4x + 5.
Solutions to Linear Systems
- A given system with equations y = 2x - 5 and -8x - 4y = -20 has one solution at the point (2.5, 0).
- The equations y = -1/2x + 4 and 2y = -8 indicate no solution as the lines are parallel.
Infinite Solutions in Linear Systems
- Infinite solutions occur when two equations represent the same line.
- An equation that could yield infinite solutions: if 3y = 2x - 9, an alternative could be y = 2/3x - 3.
No Solutions in Linear Systems
- A system of equations such as 2x - y = 7 and y = 2x + 3 results in no solution as the lines overlap without intersecting.
- Values of m and b that produce no solution must result in parallel lines, exemplified by m = -2.
Special Cases in Systems of Equations
- For the equation y = 6x + b, choosing b = -6 leads to an infinite number of solutions when paired with another compatible equation.
- Graphing system: y = -3 and y = x - 0.8 produces a solution approximated at (-2.2, -3).
Relationships Between Equations
- The equations y = -6x + 2 and -12x - 2y = -4 will yield an infinite number of solutions, indicating they are equivalent.
- A line having all the same solutions as another can be represented in different forms, such as x - 2/5y = 25/2 being equivalent to y = 2/5x - 5.
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