Find Ordered Pair That Is a Solution to the Equation

Determining Whether An Ordered Pair is a Solution of a System of Equations

Determining Whether An Ordered Pair is a Solution of a System of Equations

In Solving Linear Equations and Inequalities we learned how to solve linear equations with one variable. Remember that the solution of an equation is a value of the variable that makes a true statement when substituted into the equation.

Now we will work with systems of linear equations, two or more linear equations grouped together.

System of Linear Equations

When two or more linear equations are grouped together, they form a system of linear equations.

We will focus our work here on systems of two linear equations in two unknowns. Later, you may solve larger systems of equations.

An example of a system of two linear equations is shown below. We use a brace to show the two equations are grouped together to form a system of equations.

\(\begin{array}{c}2x+y=7\hfill \\ x-2y=6\hfill \end{array}\)

A linear equation in two variables, like 2x + y = 7, has an infinite number of solutions. Its graph is a line. Remember, every point on the line is a solution to the equation and every solution to the equation is a point on the line.

To solve a system of two linear equations, we want to find the values of the variables that are solutions to both equations. In other words, we are looking for the ordered pairs (x, y) that make both equations true. These are called the solutions to a system of equations.

Solutions of a System of Equations

Solutions of a system of equations are the values of the variables that make all the equations true. A solution of a system of two linear equations is represented by an ordered pair (x, y).

To determine if an ordered pair is a solution to a system of two equations, we substitute the values of the variables into each equation. If the ordered pair makes both equations true, it is a solution to the system.

Let's consider the system below:

\(\begin{array}{c}3x-y=7\hfill \\ x-2y=4\hfill \end{array}\)

Is the ordered pair \(\left(2,-1\right)\) a solution?

The ordered pair (2, −1) made both equations true. Therefore (2, −1) is a solution to this system.

Let's try another ordered pair. Is the ordered pair (3, 2) a solution?

The ordered pair (3, 2) made one equation true, but it made the other equation false. Since it is not a solution to both equations, it is not a solution to this system.

Example

Determine whether the ordered pair is a solution to the system: \(\begin{array}{c}x-y=-1\hfill \\ 2x-y=-5\hfill \end{array}\)

a. \(\left(-2,-1\right)\)
b. \(\left(-4,-3\right)\)

Solution

(a)

(−2, −1) does not make both equations true. (−2, −1) is not a solution.

(b)

(−4, −3) does not make both equations true. (−4, −3) is a solution.

[Attributions and Licenses]

---

• Tutorial Lessons

---

• Introduction to Systems of Linear Equations

---

• Determining Whether An Ordered Pair is a Solution of a System of Equations

---

• Solving a System of Linear Equations By Graphing

---

• Determining the Number of Solutions of a Linear System

---

• Solving Applications of Systems of Equations By Graphing

---

• Solving Systems of Equations By Graphing Key Concepts

---

• Solving a System of Equations By Substitution

---

• Solving Applications of Systems of Equations By Substitution

---

• Solving a System of Equations By Elimination

---

• Solving Applications of Systems of Equations By Elimination

---

• Choosing the Most Convenient Method to Solve a System of Linear Equations

---

• Solving Applications With Systems of Equations

---

• Translating to a System of Equations

---

• Solving Direct Translation Applications

---

• Solving Geometry Applications

---

• Solving Uniform Motion Applications

---

• Solving Mixture Applications

---

• Solving Interest Applications

---

• Determining Whether An Ordered Pair is a Solution of a System of Linear Inequalities

---

• Solving a System of Linear Inequalities By Graphing

---

• Solving Applications of Systems of Inequalities

---

• Graphing Systems of Linear Inequalities Summary

---

Find Ordered Pair That Is a Solution to the Equation

Source: https://nigerianscholars.com/tutorials/systems-of-linear-equations/determining-whether-an-ordered-pair-is-a-solution-of-a-system-of-equations/

Share this post