Problem 33 - Algebra 1

Question

Use linear systems to determine where the \(x\) -coordinate is equal to the \(y\) -coordinate on the graph of \(2 x+3 y=25\).

Step-by-Step Solution

Answer

The point on the graph of \(2x + 3y = 25\) where the x-coordinate equals the y-coordinate is (5,5).

Step 1: Specify the Coordinates

The coordinates that we need to find are where the x and y are equal. Let's represent these equal coordinates as \(a\). Therefore the coordinates become \((a, a)\).

Step 2: Substitute into the Equation

Substitute \(a\) for \(x\) and \(y\) in the equation \(2x + 3y = 25\) to derive a new equation. This results in the equation \(2a + 3a = 25\).

Step 3: Solve for 'a'

Combine like terms to get \(5a = 25\). Then divide by 5 to solve for \(a\), so \(a = 5\). Therefore, the coordinates (x,y) on the line where x=y are (5,5).

Key Concepts

Coordinate Geometry

In coordinate geometry, we explore the properties and relationships of points, lines, and shapes on a plane using a coordinate system. It's a way to connect algebra and geometry, providing a visual understanding of mathematical concepts.

When dealing with linear equations, such as the one in this problem, we are often trying to find out where two lines intersect or specific points on a line. Each point on the plane is described using coordinates \(x, y\).

In this exercise, we're looking for the point where the \(x\)-coordinate is equal to the \(y\)-coordinate. This translates to finding the intersection point on the line \(2x + 3y = 25\) where these coordinates are the same i.e., \(x = y\). By applying a systematic approach to coordinate geometry, we can determine this intersection point, which visually represents where the line crosses the line \(y = x\).

Linear Equations

Linear equations form the backbone of many mathematical concepts, representing straight lines on the coordinate plane. A linear equation like \(2x + 3y = 25\) is an equation of the first degree, meaning it does not contain variables raised to powers other than one.

Here are some key properties of linear equations:

• It graphs as a straight line.
• The equation can be written in various forms, such as slope-intercept form \(y = mx + b\) or standard form \(Ax + By = C\).
• It has a constant rate of change, or slope, which represents the steepness of the line.

Our task is to find specific points on this line. By setting \(x = y\), we adapt the equation to find where these coordinates coincide. This transformation simplifies solving the problem by focusing on one variable representing both \(x\) and \(y\).

Substitution Method

The substitution method is a classic algebraic technique used to solve systems of equations. It's particularly effective when one of the equations already defines a variable in terms of the other.

Here’s how it works in our context:

• Identify the condition \(x = y\) and represent them as a single variable \(a\): \(x = a\) and \(y = a\).
• Substitute \(a\) into the given equation \(2x + 3y = 25\) to form a single equation: \(2a + 3a = 25\).
• Simplify the equation to find the value of \(a\).

This method allows us to reduce a two-variable problem into a one-variable equation, making it easier to solve. The solution \(a = 5\) tells us that the coordinates (5,5) satisfy the original equation under the condition \(x = y\). This demonstrates the power of the substitution method in simplifying complex equations.