Problem 49 - Algebra 1: Concepts and Skills

Question

Determine whether the ordered pair is a solution of the equation. (Lesson 4.2) $$ x-y=10,(5,-5) $$

Step-by-Step Solution

Answer

Yes, the ordered pair \((5,-5)\) is a solution to the equation \(x-y=10\).

Step 1: Substitute the values of the ordered pair into the equation

In the ordered pair \((5,-5)\), '5' is the 'x' coordinate and '-5' is the 'y' coordinate. Now, substitute these values into the equation \(x-y=10\), it becomes \(5-(-5) = 10\).

Step 2: Simplify the equation

The equation \(5-(-5) = 10\) simplifies to \(5 + 5 = 10\). The left hand side of this equation simplifies further to '10'.

Step 3: Check if both sides of the equation are equal

Now it needs to be checked whether the simplified left hand side is equal to the right hand side of the equation. '10' is equal to '10', which indeed it is. Therefore, the ordered pair \((5,-5)\) is a solution to the equation \(x-y=10\).

Key Concepts

Ordered Pairs

An ordered pair is a fundamental element in algebra and coordinate geometry, represented by two numbers in a specific, fixed order. The first number is the x-coordinate, and the second number is the y-coordinate. Ordered pairs are written in the form \( (x, y) \), and they pinpoint a precise location on a two-dimensional plane, commonly known as the coordinate plane.

In our exercise, the ordered pair \( (5, -5) \) is given, which suggests that we are looking at a point located at 5 units along the x-axis and -5 units along the y-axis. Understanding ordered pairs is critical for solving equations since they allow us to test whether a certain point lies on the line or curve defined by an equation.

Substitution Method

The substitution method is a technique used to solve systems of equations or to check the validity of potential solutions. This method involves replacing variables in an equation with their corresponding values to simplify and solve it.

In our textbook exercise, we applied this method by substituting the x and y coordinates of the ordered pair \( (5, -5) \) into the equation \( x-y=10 \). Substituting '5' for x and '-5' for y gives us the expression \( 5-(-5)=10 \). By performing the mathematical operations, we simplified it to \( 10=10 \), which confirms that the values match and hence, \( (5, -5) \) is indeed a solution to the equation.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface where points are placed using ordered pairs \( (x, y) \). The plane is divided into four quadrants by two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical). Points on the plane are referenced in relation to these axes.

The importance of the coordinate plane lies in its ability to visually represent algebraic equations. For example, the equation \( x-y=10 \) from our exercise represents a straight line on this plane. Each solution to the equation, including ordered pairs like \( (5, -5) \) that satisfy it, can be seen as points on this line. Understanding the coordinate plane is essential for graphing equations and interpreting their solutions in a visual context.