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.