Problem 30 - Prealgebra

Question

Indicate which of the given ordered pairs are solutions for each equation. $$3 x+7 y=21 \quad(0,3),(7,0),(1,2)$$

Step-by-Step Solution

Answer

The solutions are (0, 3) and (7, 0).

Step 1: Substitute the First Pair

Substitute the first pair (0, 3) into the equation \(3x + 7y = 21\). We find \(3(0) + 7(3) = 0 + 21 = 21\). Since the left side equals the right side, (0, 3) is a solution.

Step 2: Substitute the Second Pair

Substitute the second pair (7, 0) into the equation \(3x + 7y = 21\). We find \(3(7) + 7(0) = 21 + 0 = 21\). Since the left side equals the right side, (7, 0) is a solution.

Step 3: Substitute the Third Pair

Substitute the third pair (1, 2) into the equation \(3x + 7y = 21\). We find \(3(1) + 7(2) = 3 + 14 = 17\). Since 17 does not equal 21, (1, 2) is not a solution.

Key Concepts

Ordered Pairs

When dealing with linear equations, ordered pairs are crucial for determining solutions. An ordered pair, typically written as \((x, y)\), represents a point on the Cartesian coordinate plane, where \(x\) is the horizontal component, and \(y\) is the vertical component. In the context of equations, these pairs allow us to substitute values directly into the expression.
Understanding how to substitute and check these pairs helps in verifying if they satisfy the given equation. For instance, to check if an ordered pair like \((0, 3)\) is a solution to the equation \(3x + 7y = 21\), you substitute \(x = 0\) and \(y = 3\) into the equation.

• If the resulting equation is true, the ordered pair is a solution.
• If it is false, the ordered pair is not a solution.

Taking time to verify ordered pairs helps ensure the accuracy of solving equations.

Solutions of Equations

Linear equations can have infinite, one, or no solutions when plotted on a graph. A solution of a linear equation is an ordered pair that makes the equation true when the values are substituted back into it.
Using our example, the equation \(3x + 7y = 21\) is satisfied by certain points, known as solutions:

• Substitute values from an ordered pair into the equation.
• Simplify to see if both sides of the equation are equal.

For instance, when substituting \((7, 0)\) into the equation, both sides become equal, confirming it as a solution. However, some pairs like \((1, 2)\) when substituted do not equate, indicating that it is not a solution. This verification is vital when working with systems of equations or graphing lines.

Substitution Method

The substitution method is a valuable algebraic technique used to find solutions of linear equations, particularly in systems of equations. It involves substituting values to check their validity in a given equation.
In our example, the substitution method was used to verify ordered pairs. Here's how it works:

• Choose an ordered pair to test.
• Substitute these values into the equation, replacing \(x\) and \(y\) with their respective numbers.
• Simplify the equation to check if both sides are equivalent.

This method simplifies the process of verifying whether a specific ordered pair is a solution to the equation, ensuring you accurately determine which points lie on the line represented by the equation.