Questions and Answers
Which values of m and b will create a system of equations with no solution? Select two options.
How many solutions does the linear system y = 2x - 5 and -8x - 4y = -20 have?
(2.5, 0)
If 4x - y = 5 is one of the equations, which could be the other equation if the system has 1 solution?
y = -4x + 5
What is the solution to the system of linear equations?
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Which equation could be Henry's if his equation had all the same solutions as Fiona's equation y = 2/5x - 5?
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If y = 8x + 7 is one of the equations and the system has no solution, which could be the other equation?
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What could be the other equation if one of the equations of Muriel's system is 3y = 2x - 9, which has an infinite number of solutions?
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How many solutions does the linear system y = 1/2x + 4 and x + 2y = -8 have?
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What value of b will cause the system to have an infinite number of solutions for y = 6x + b and -3x + 1/2y = -3?
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Which is the best approximate solution of the system of linear equations y = 1.5x - 1 and y = 1?
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Study Notes
Systems of Linear Equations: Key Concepts
- A system of equations can have no solution if the lines are parallel, indicated by having the same slope (m) but different y-intercepts (b).
- For example, equations y = -2x + 3/2 and lines with m = -2 and various b values result in no solution if values are different.
Solution Characteristics
- A linear system can possess one solution, signifying the intersection of two lines at a specific point, such as (2.5, 0) for the system involving y = 2x - 5 and -8x - 4y = -20.
- To have one solution, the equations must not be identical and must intersect at one point.
Infinite Solutions
- A linear system has infinite solutions when both equations represent the same line. For instance, if one equation is 4x - y = 5, a possible corresponding equation could be y = -4x + 5.
Specific Solutions
- The solution to a system of linear equations can be represented as a coordinate point. For example, (0, 2) may be the solution for a given set of equations.
Equivalent Equations
- Two equations can have identical solutions. For example, if Fiona's equation is y = 2/5x - 5, Henry's equivalent could be x - 5/2y = 25/2, maintaining the same solution set.
No Solutions
- A system can have no solutions if the equations are contradictory. For example, y = 8x + 7 has a corresponding equation, y = 8x - 7, showcasing no intersection.
Infinite Solutions Example
- An infinite number of solutions occurs when one equation can be derived from manipulating another. For instance, with 3y = 2x - 9, a compatible equation is y = 2/3x - 3.
Analyzing Solutions
- Sometimes a system of equations can clearly indicate no solutions, such as y = 1/2x + 4 and x + 2y = -8, which are parallel.
Impact of b on Solutions
- The value of b can affect the number of solutions in a system. For instance, when y = 6x + b, setting b to -6 results in infinite solutions with the equation -3x + 1/2y = -3.
Finding Approximate Solutions
- Approximate solutions can be determined graphically or through calculation, such as the solution (1.83, 1) for the equations y = 1.5x - 1 and y = 1.
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Description
This set of flashcards focuses on solving systems of linear equations through graphing techniques. It includes various scenarios such as identifying when there are no solutions and determining the number of solutions for given equations. Test your understanding and improve your skills in interpreting and solving linear models.
