Identifying Solutions to a Linear Equation in Two Variables

Fundamental Principles of Linear Equations in Two Variables

• Definition of a Linear Equation in Two Variables:

  • A linear equation in two variables is an algebraic equation that can be written in standard form:     $Ax + By = C$
  • In this expression, $A$, $B$, and $C$ are real numbers, where $A$ and $B$ are not both equal to zero.
  • The variables $x$ and $y$ represent unknown quantities or coordinates on a two-dimensional Cartesian plane.

• Definition of an Ordered Pair:

  • An ordered pair is written in the standard coordinate format $(x, y)$.
  • The first value, $x$, represents the input or horizontal position.
  • The second value, $y$, represents the output or vertical position.

• Definition of a Solution:

  • An ordered pair $(x, y)$ is a solution to a two-variable equation if substituting the corresponding values into the equation yields a true mathematical statement (where the left-hand side equals the right-hand side).
  • If substitution results in an inequality (where the left-hand side does not equal the right-hand side), the ordered pair is not a solution.

Systematic Method for Evaluating Ordered Pairs

• Step 1: Identify the given linear equation:   $7x - 6y = 19$

• Step 2: Extract the numerical values for $x$ and $y$ from the given ordered pair $(x, y)$.

• Step 3: Substitute the value of $x$ and the value of $y$ into the expression on the left-hand side of the equation:   $7x - 6y$

• Step 4: Evaluate the expression using standard order of operations (multiplication before subtraction/addition):

  • Multiply the coefficient $7$ by the $x$-coordinate.
  • Multiply the coefficient $6$ by the $y$-coordinate.
  • Subtract the second product from the first product, paying close attention to sign rules for negative numbers:
  • Positive multiplied by negative equals negative.
  • Subtracting a negative number is equivalent to adding its positive value.

• Step 5: Compare the evaluated result with the constant on the right-hand side ($19$):

  • If the simplified left-hand side equals $19$, the ordered pair is a solution.
  • If the simplified left-hand side does not equal $19$, the ordered pair is not a solution.

Detailed Evaluation of Each Ordered Pair

• Evaluation of $(7, 5)$:

  • Value of $x$: $7$
  • Value of $y$: $5$
  • Substitution into the expression $7x - 6y$:     $7(7) - 6(5)$
  • Perform multiplications:     $49 - 30$
  • Perform subtraction:     $19$
  • Compare with target value:     $19 = 19$
  • Conclusion: The equation holds true. Therefore, $(7, 5)$ is a solution (Yes).

• Evaluation of $(-4, 0)$:

  • Value of $x$: $-4$
  • Value of $y$: $0$
  • Substitution into the expression $7x - 6y$:     $7(-4) - 6(0)$
  • Perform multiplications:     $-28 - 0$
  • Perform subtraction:     $-28$
  • Compare with target value:     $-28 \neq 19$
  • Conclusion: The equation is false. Therefore, $(-4, 0)$ is not a solution (No).

• Evaluation of $(2, -1)$:

  • Value of $x$: $2$
  • Value of $y$: $-1$
  • Substitution into the expression $7x - 6y$:     $7(2) - 6(-1)$
  • Perform multiplications:     $14 - (-6)$
  • Apply double negative rule:     $14 + 6$
  • Perform addition:     $20$
  • Compare with target value:     $20 \neq 19$
  • Conclusion: The equation is false. Therefore, $(2, -1)$ is not a solution (No).

• Evaluation of $(-5, -3)$:

  • Value of $x$: $-5$
  • Value of $y$: $-3$
  • Substitution into the expression $7x - 6y$:     $7(-5) - 6(-3)$
  • Perform multiplications:     $-35 - (-18)$
  • Apply double negative rule:     $-35 + 18$
  • Perform addition:     $-17$
  • Compare with target value:     $-17 \neq 19$
  • Conclusion: The equation is false. Therefore, $(-5, -3)$ is not a solution (No).

Solution Summary Table

• Complete Verification Results for Equation $7x - 6y = 19$:
  • Ordered Pair $(7, 5)$: Calculated Value = $19$ | Solution Status = Yes
  • Ordered Pair $(-4, 0)$: Calculated Value = $-28$ | Solution Status = No
  • Ordered Pair $(2, -1)$: Calculated Value = $20$ | Solution Status = No
  • Ordered Pair $(-5, -3)$: Calculated Value = $-17$ | Solution Status = No