Question

Describe the equations of the linear functions that go through the point \((2,5) .\) Give 2 examples.

Short answer

The linear equations are \(y = x + 3\) and \(y = -2x + 9\).

Step-by-step solution

Understand the Problem

We need to find the equations of linear functions that pass through the point \((2, 5)\). A linear function is generally in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. The point \((2, 5)\) must satisfy the equation.

Use the Point to Formulate Equations

For a line with an unknown slope \(m\), passing through the point \((2, 5)\), we use the point-slope form of the equation: \(y - y_1 = m(x - x_1)\). Substitute \(x_1 = 2\) and \(y_1 = 5\) into the equation: \(y - 5 = m(x - 2)\).

Solve for the General Equation

Rearrange the equation \(y - 5 = m(x - 2)\) to express \(y\) explicitly: \(y = m(x - 2) + 5\). Simplify it to \(y = mx - 2m + 5\). Here, \(b = -2m + 5\).

Select a Specific Slope Example 1

Let's choose \(m = 1\) as an example slope. Substituting \(m = 1\) in \(y = mx - 2m + 5\): \ \(y = 1x - 2 \times 1 + 5\), which simplifies to \(y = x + 3\). This is the first linear equation.

Select a Specific Slope Example 2

Choose another slope, \(m = -2\), for variety. Substitute \(m = -2\) in \(y = mx - 2m + 5\): \ \(y = -2x - 2 \times (-2) + 5\). Simplifying this gives \(y = -2x + 9\). This is the second linear equation.

Key concepts

Point-Slope Form

The point-slope form is an incredibly useful tool for writing the equation of a line when you have a point on the line and the slope. The form is written as follows:

• \(y - y_1 = m(x - x_1)\)

In this equation, \((x_1, y_1)\) represents a specific point on the line, and \(m\) is the slope. By substituting the given point into the equation, you can create a new equation that describes all the points on that line.
To make an equation using the point-slope form, you:

• Identify a known point on the line (in this case, \((2, 5)\)).
• Utilize a specific slope value that the line will possess.

By plugging these values into the point-slope form, you derive a linear equation that describes the line. It's a straightforward method that offers flexibility with various slopes.

Slope-Intercept Form

The slope-intercept form is another popular way of writing the equation of a line. This form closely resembles the format mentioned in linear functions and is expressed as:

• \(y = mx + b\)

Here, \(m\) represents the slope, and \(b\) is the y-intercept, or the point where the line crosses the y-axis.
The slope-intercept form is particularly useful because it immediately tells you both the slope and the starting position of the line on the graph. To convert an equation from point-slope form to slope-intercept form, as the example problem required:

• Rearrange the point-slope equation to solve for \(y\).
• Simplify to fit the \(y = mx + b\) structure.

This transition makes it evident how different slope choices affect the line's behavior while ensuring it still passes through the specified point.

Linear Functions

Linear functions represent the simplest form of functions in mathematics and are expressed as a straight line when graphed. Each linear function can be described using the equation:

• \(y = mx + b\)

where \(m\) is the slope and \(b\) is the y-intercept.
Linear functions display a constant rate of change and are powerful tools for representing relationships in data and real-world scenarios. They are the backbone of predictive modeling in both mathematics and science.
Whether you're using the point-slope form or slope-intercept form, linear functions are all about forming straight lines, making them easily predictable and interpretable. When working through exercises that involve finding linear function equations, identifying the key components such as points and slopes helps you build these representations.

Algebraic Expressions

In algebra, expressions are combinations of numbers, variables, and operators. When solving problems involving lines, you'll often rearrange these expressions to simplify or identify specific components, such as solving for the slope or intercepts of a line.

• An algebraic expression can range from a simple \(x + 3\) to more complex formulations like \(2x^2 - 4x + 7\).

In the realm of linear equations, you manipulate these expressions to convert between different forms of the line's equation.
Whether you're working on point-slope form or slope-intercept form, understanding how to rearrange and simplify algebraic expressions is vital. It's not just about numbers and variables; it's about unlocking the relationships they describe.
Simplifying algebraic expressions to reach desired forms helps illuminate the properties of the line, such as finding \(y = x + 3\) or \(y = -2x + 9\) from the original examples.