Problem 43 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether each ordered pair is a solution of the given equation. $$3 x+5 y=15 \quad(-5,6),(0,5),(10,-3)$$

Step-by-Step Solution

Answer

The ordered pairs (-5,6) and (10,-3) are solutions to the equation, while the ordered pair (0,5) is not a solution.

Step 1: Substitute the Ordered Pair (-5,6) into the Equation

The ordered pair (-5,6) means that \(x=-5\) and \(y=6\). Substitute these into the equation: \(3*(-5)+5*6=-15+30=15\). This is equal to the right-hand side of the equation.

Step 2: Substitute the Ordered Pair (0,5) into the Equation

The ordered pair (0,5) means that \(x=0\) and \(y=5\). Substitute these into the equation: \(3*0+5*5=0+25=25\). This is not equal to the right-hand side of the equation.

Step 3: Substitute the Ordered Pair (10,-3) into the Equation

The ordered pair (10,-3) means that \(x=10\) and \(y=-3\). Substitute these into the equation: \(3*10+5*(-3)=30-15=15\). This is equal to the right-hand side of the equation.

Key Concepts

Ordered Pairs

In the realm of mathematics, ordered pairs are fundamental to understanding coordinate geometry and solving linear equations. Ordered pairs, written as \( (x, y) \), represent the coordinates of a point on a Cartesian plane where \( x \) is the horizontal value and \( y \) is the vertical value. These pairs help in visualizing mathematical relations and functions.
When dealing with linear equations, ordered pairs are actually potential solutions. By plugging the values of an ordered pair into the equation, we can verify its validity as a solution. If upon substituting the ordered pair, the equation balances, then that pair is indeed a solution to the equation.

Substitution Method

The substitution method is an algebraic technique used to solve systems of equations. In this method, we express one variable in terms of the other, and then substitute that expression into another equation. However, the substitution method can also be applied in a simpler form when investigating if an ordered pair is a solution to an equation.
When examining ordered pairs as potential solutions, this method involves replacing the variables with the respective values from the pair. For instance, if we have the ordered pair \( (a, b) \) and want to check if it is a solution to the equation \( cx + dy = e \), we would substitute \( x \) with \( a \) and \( y \) with \( b \) and simplify. If the equation holds true, then the ordered pair \( (a, b) \) is indeed a solution.

Linear Equation Solutions

Solving a linear equation means finding all the values that make the equation true. The solutions to a linear equation are the set of all ordered pairs that, when substituted into the equation, produce a true statement. The most common way that linear equations are presented is in the form \( ax + by = c \), where \( a \), \( b \) and \( c \) are constants and \( x \) and \( y \) are the variables that represent coordinates on a plane.
In the context of the provided exercise, an ordered pair is a solution if, after substitution, the left-hand side of the equation has the same value as the constant on the right-hand side. Therefore, the concept of linear equation solutions is centered around the search for these pairs that satisfy the given equation.

Algebraic Operations

To work effectively with linear equations and ordered pairs, one must be proficient with algebraic operations. These include addition, subtraction, multiplication, and division of numerical values and variables. In the process of checking solutions for a linear equation, these operations are used extensively.
For example, to substitute an ordered pair into a linear equation, you may need to multiply the \( x \) value by its coefficient and do the same for the \( y \) value before adding or subtracting these results as required by the equation. This precise use of algebraic operations is crucial when assessing whether a given ordered pair solves the equation.