Solving Systems of Linear Equations Flashcards

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.