Solving Systems of Linear Equations Flashcards

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Questions and Answers

Which values of m and b will create a system of equations with no solution? Select two options.

  • m = -2 and b = -1/3 (correct)
  • m = -2 and b = -2/3 (correct)
  • m = -1 and b = 0
  • m = 2 and b = 5

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?

<p>(0, 2)</p> Signup and view all the answers

Which equation could be Henry's if his equation had all the same solutions as Fiona's equation y = 2/5x - 5?

<p>x - 5/2y = 25/2</p> Signup and view all the answers

If y = 8x + 7 is one of the equations and the system has no solution, which could be the other equation?

<p>y = 8x - 7</p> Signup and view all the answers

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?

<p>y = 2/3x - 3</p> Signup and view all the answers

How many solutions does the linear system y = 1/2x + 4 and x + 2y = -8 have?

<p>no solution</p> Signup and view all the answers

What value of b will cause the system to have an infinite number of solutions for y = 6x + b and -3x + 1/2y = -3?

<p>-6</p> Signup and view all the answers

Which is the best approximate solution of the system of linear equations y = 1.5x - 1 and y = 1?

<p>(1.83, 1)</p> Signup and view all the answers

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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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