Problem 19 - Algebra 1: Concepts and Skills

Question

Determine whether the ordered pair is a solution of the equation. $$ 5 x-8 y=15,(3,0) $$

Step-by-Step Solution

Answer

Yes, the ordered pair (3,0) is a solution to the equation \(5x - 8y = 15\)

Step 1: Substitute the Ordered Pair

Substitute the ordered pair (3,0) into the equation. Replace \(x\) with 3 and \(y\) with 0 in the equation to yield \(5 * 3 - 8 * 0\).

Step 2: Perform Operations

Carry out the multiplication and subtraction operations. The left side of the equation becomes \(5 * 3 = 15\), and since there is no \(y\) term due to it being multiplied by 0, that yields \(15\).

Step 3: Compare the Result with the Right Side of the Equation

The right side of the equation is also \(15\). Since both the left and right side of the equation give the same result, the ordered pair (3,0) is a solution to the equation.

Key Concepts

Ordered Pair

An ordered pair is a fundamental concept in mathematics, particularly in coordinate geometry. It consists of two numbers written in a specific sequence, usually presented as \((x, y)\). The first value represents the horizontal position (x-coordinate), and the second value represents the vertical position (y-coordinate).
An ordered pair can be used to denote a specific point on a coordinate plane, which consists of two axes: the horizontal axis (x-axis) and the vertical axis (y-axis).
Understanding ordered pairs is crucial because they provide a clear way of pinpointing locations and describing relationships between variables.

• Example: In the ordered pair \((3, 0)\), \(3\) is the x-coordinate and \(0\) is the y-coordinate.
• Ordered pairs are typically used in various mathematical contexts, such as graphing equations and describing solutions of equations.

When determining if an ordered pair is a solution to an equation, you substitute them into the equation to verify if they satisfy the equation.

Linear Equations

Linear equations are equations of the first degree, meaning they have no exponents greater than one. They typically have a format like \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. Linear equations can describe straightforward relationships between two variables.
These equations graph as straight lines on a coordinate plane, hence the name "linear." The key aspects of linear equations include:

• The "constant term" \(c\) is where the line intersects the y-axis if the equation is of the form \(y = mx + c\).
• "m" represents the slope, which indicates the steepness of the line and the direction it goes.
• Linear equations are fundamental in modeling real-world situations.

When solving a linear equation, you're often finding the value of x and y that makes the equation true. This can involve checking if an ordered pair lies on the line represented by the linear equation.

Substitution Method

The substitution method is a powerful technique used in algebra to solve a system of equations. Although the given exercise involves only one equation, understanding substitution helps when dealing with multiple equations.
This method involves replacing one variable with an expression to simplify and solve the equation. Here's a simple way to think about it:

• Choose one of the equations and solve it for one variable in terms of the other.
• Substitute this expression into the other equation. This substitution should now allow you to solve for a single variable.
• Once that variable is found, replace it back into one of the original equations to solve for the other variable.

In the context of the exercise provided, we substituted \(x = 3\) and \(y = 0\) directly into the equation and checked if both sides matched. The substitution method primarily aids in determining the validity of solutions like ordered pairs.