Study Notes on Solving Systems of Linear Equations

SOLVING SYSTEMS OF LINEAR EQUATIONS BY GRAPHING, SUBSTITUTION, AND ELIMINATION

Different Methods for Solving Linear Equations

• Graphing: Involves plotting each line and identifying the intersection point.

• Substitution: Involves solving one equation for a single variable and substituting it into the other equation.

• Elimination: Involves manipulating the equations to eliminate one variable, making it easier to solve for the other.

GRAPHING

• Steps to Graph Each Line:

  • Write each equation in slope-intercept form, y = mx + b where (m) is the slope and (b) is the y-intercept.

  • For example, break down the given equations:

  • From the equation x + 2y = 4:

    • Rearranging gives:

    • 2y = -x + 4

    • y = - rac{1}{2}x + 2

  • From the equation 3x = 5 + y:

    • Rearranging gives:

    • y = 3x - 5

  • Graphing the Equations:

  • Plot the lines on the same coordinate plane and find the intersection point.

    • The intersection point for the equations plotted will be (2, 1), meaning the solution is:

    • Solution: The point of intersection is (2, 1).

SUBSTITUTION

• Process of Substitution:

  • Choose one equation to solve for one of the variables:

  • For example, solving the first equation for (x):

    • Starting from x + 2y = 4:

    • x + 2y - 2y = 4 - 2y

    • x = 4 - 2y

  • Substitute this expression for (x) into the second equation:

  • For the second equation, 3x = 5 + y:

    • Substituting yields:

    • 3(4 - 2y) = 5 + y

    • Simplifying results in:

      • 12 - 6y = 5 + y

  • Rearranging and solving for (y):

  • 12 - 6y + 6y = 5 + y + 6y

  • 12 = 5 + 7y

  • Subtracting results in:

    • 12 - 5 = 7y

    • 7 = 7y

    • y = 1

  • Substitute (y = 1) back into any equation to solve for (x):

  • Plugging into x + 2y = 4 results in:

    • x + 2(1) = 4

    • x + 2 = 4

    • x = 2

  • Final Solution: The solution through substitution also indicates: (2, 1).

ELIMINATION

• Method of Elimination:

  • Rearrange equations to align like terms for addition or subtraction to eliminate a variable:

  • From the equation 3x = 5 + y, rearranging gives:

    • 3x - y = 5

  • To eliminate (y), adjust coefficients:

  • Multiply the second equation by 2:

    • 2(3x - y) = 2(5) resulting in:

    • 6x - 2y = 10

  • Add the Equations:

  • Aligning the equations:

    • x + 2y = 4

    • + 6x - 2y = 10

    • Yields:

    • 7x + 0y = 14

  • Solving for (x):

  • Dividing through yields:

    • 7x = 14

    • x = 2

  • Substitute (x = 2) back into either equation to solve for (y):

  • Using x + 2y = 4 results in:

    • 2 + 2y = 4

    • 2y = 4 - 2

    • 2y = 2

    • y = 1

  • Final Solution: The solution through elimination also confirms: (2, 1).

SYSTEMS OF EQUATIONS: NUMBER OF SOLUTIONS

• A solution to a system of linear equations represents the point of intersection.

• There are three possibilities for the number of solutions:

  1. One solution: Occurs when the lines intersect at exactly one point, implying the slopes are different.

  • Example: System:

    • y = -2x + 4

  1. No solution: This occurs if lines are parallel, characterized by the same slope but different y-intercepts.

  • Example:

    • System:

      • y = 2x + 4 and y = 2x - 3

      • The lines do not intersect indicating no solution.

  1. Infinitely many solutions: This occurs if the equations represent the same line.

  • Example:

    • System:

    • y = -4x + 3 and 12x + 3y = 9

    • Rearranging the second equation yields:

      • 3y = -12x + 9

      • y = -4x + 3

    • Since both equations represent the same line, there are infinitely many solutions.

Practice Problems

• Determine if each given system has:

  • One solution

  • No solution

  • Infinitely many solutions

  • Following completion, explain the reasoning behind each conclusion by analyzing the slopes and intercepts of the equations provided.