Problem 70 - Algebra 1: Concepts and Skills

Question

Determine whether the ordered pair is a solution of the system of linear equations. $$ \begin{aligned} &2 x+6 y=22\\\ &-x-4 y=-13 \quad(-5,-2) \end{aligned} $$

Step-by-Step Solution

Answer

The ordered pair (-5, -2) is a solution to the system of equations as it satisfies both equations.

Step 1: Substitute into the First Equation

Substitute the numbers -5 and -2 for x and y in the first equation, i.e., \(2*(-5) + 6*(-2)\).

Step 2: Simplify and Evaluate

Simplify the expression to get \(-10 - 12 = -22\)

Step 3: Substitute into the Second Equation

Substitute the numbers -5 and -2 for x and y in the second equation, i.e., \(-(-5) - 4*(-2)\).

Step 4: Simplify and Evaluate

Simplify the expression to get \(5 + 8 = 13\).

Key Concepts

Ordered Pairs

An ordered pair is a fundamental concept in coordinate geometry and algebra. It consists of two values, typically written in the form

• (x, y)

where the first value corresponds to the x-coordinate and the second value corresponds to the y-coordinate.
Ordered pairs are used to define specific points on a two-dimensional plane, making them essential in graphing and solving mathematical equations.
To determine whether an ordered pair satisfies a given equation, you substitute the values of the pair into the equation and check if the equation holds true.
For instance, in the ordered pair (-5, -2), -5 is the x-coordinate and -2 is the y-coordinate. When checking this ordered pair against a system of equations, substitute these values into each equation to see if they result in true statements.
This involves plugging in the x and y values into the equations to verify the validity of the pair as a solution.
This substitution process will be akin to a test where correct results indicate a verified solution.

Systems of Linear Equations

A system of linear equations comprises two or more linear equations that involve the same set of variables.
Linear equations appear in the standard form as

• \(ax + by = c\)

where a, b, and c are constants, and x and y are variables.
The goal is to find a set of values for these variables that satisfies all equations simultaneously.
Each linear equation represents a straight line when graphed on a coordinate plane. For a system of equations to have a solution, these lines must intersect at a common point.
This point of intersection is represented by an ordered pair which is the solution to the system.
In example systems:

• \(2x+6y=22\)
  }
• \(-x-4y=-13\)

To find whether a given ordered pair, say (-5, -2), is a solution, you substitute it into each equation and check for consistency.
Thus, solving a system of linear equations is about finding common solutions or verifying if a proposed solution works for all equations in the system.

Solution Verification

Solution verification is a vital step in ensuring that an ordered pair is indeed a solution to a system of equations.
To verify a solution, substitute the values of the ordered pair into each equation of the system, and ensure the equation is satisfied.
This involves checking both sides of each equation to confirm they are equal after substitution of the ordered pair.
For instance, consider verifying whether (-5, -2) satisfies the system:

• \(2x + 6y = 22\)
• \(-x - 4y = -13\)

- **Substitute (-5) for x and (-2) for y in the first equation:** \[2(-5) + 6(-2)\] Simplify to get \[-10 - 12 = -22\], which is not equal to 22, so the equation is not satisfied.- **Substitute in the second equation:** \[-(-5) - 4(-2)\] Simplify to get \[5 + 8 = 13\], which is equal to 13, but only this step is fulfilled.Since both equations must agree for the ordered pair to be a solution, and the first was not satisfied, (-5, -2) is not a solution for the system.
This highlights the importance of verifying against each equation in the system for a thorough and complete solution check.