Problem 38 - Algebra A Combined Function

Question

Without solving, how can you tell that the graphs of \(y=2 x+3\) and \(y=2 x+7\) do not have any points of intersection?

Step-by-Step Solution

Answer

The lines are parallel with equal slopes (2) and different y-intercepts (3 and 7), so they do not intersect.

Step 1: Identify the Slope of Each Line

Both given equations, \(y=2x+3\) and \(y=2x+7\), can be identified as linear equations in the form \(y=mx+b\). Here, \(m\) represents the slope of the line. For both equations, the slope \(m\) is 2.

Step 2: Determine if Slopes Indicate Parallel Lines

Since both lines have the same slope (\(m=2\)), they are parallel to each other. Parallel lines in the same plane never intersect unless they are coincident, meaning they are the same line.

Step 3: Check the Y-Intercepts

The y-intercepts of the lines are different: one line intercepts the y-axis at 3, \(y=2x+3\), and the other at 7, \(y=2x+7\). Since the y-intercepts are different, the lines are not the same but distinct parallel lines.

Key Concepts

Linear Equations

Linear equations are algebraic expressions that describe a straight line on a graph. They are typically presented in the slope-intercept form:

• \( y = mx + b \)

where \( m \) is the slope and \( b \) is the y-intercept.
Linear equations can model countless scenarios where relationships between variables are linear. This simplicity makes them vital in mathematics and practical applications like physics and economics. Understanding linear equations enables us to predict and interpret data trends and solve various real-world problems effectively. Each equation maps to a set of points that form a straight line, symbolizing a direct proportionality between the variables involved.

Slope

The slope of a line is a measure of its steepness or the rate of change of the y-variable with respect to the x-variable. Mathematically, the slope (\( m \)) is calculated using:

• \( m = \frac{\Delta y}{\Delta x} \)

This formula finds the vertical change (rise) divided by the horizontal change (run) between two points on a line.
For the equations given, \( y = 2x + 3 \) and \( y = 2x + 7 \), the slope is 2 for both, meaning for every unit increase in x, y increases by 2 units in both lines. If two lines share the same slope, they are parallel and run alongside each other. Slope is crucial for determining the orientation and direction of a line, helping to quickly assess whether lines are parallel, perpendicular, or neither.

Y-Intercept

The y-intercept is the point at which a line crosses the y-axis. It is represented by the \( b \) in the linear equation format \( y = mx + b \). This value indicates where the line begins on the y-axis when x is zero.

• For \( y = 2x + 3, \) the y-intercept is 3.
• For \( y = 2x + 7, \) the y-intercept is 7.

The y-intercept offers an initial value to predict further points on a line, highlighting the vertical positioning of the line relative to other lines in the graph. In the case of parallel lines with identical slopes, varying y-intercepts confirm that the lines are distinct and do not coincide. Thus, different y-intercepts in parallel lines mean they will never intersect.

Intersection of Lines

The intersection of lines is the point where two lines cross each other on a graph. In mathematical terms, it refers to a point that satisfies both equations simultaneously. However, lines that share the same slope but different y-intercepts are parallel, and therefore, they do not meet at any point in the coordinate plane.
In our exercise involving the equations \( y = 2x + 3 \)and\( y = 2x + 7,\) the lines are parallel due to equal slopes but differing y-intercepts. This results in no intersection, as parallel lines maintain a consistent distance between them and continue infinitely without touching each other. Understanding the intersection, or lack thereof, becomes key in applications such as solving systems of equations, which help us find solutions where different conditions align concurrently.