Problem 47 - Algebra 1

Question

Decide whether the graphs of the two equations are parallel lines. Explain your answer. $$ 2 x-12=y, y=10+2 x $$

Step-by-Step Solution

Answer

Yes, the graphs of the two given equations form parallel lines because they have the same slope.

Step 1: Rearrange Equation 1 Into Slope-Intercept Form

Rewrite the first equation, \(2x - 12 = y\), in slope-intercept form. This is done by switching the sides, resulting in the equation \(y = 2x - 12\).

Step 2: Rearrange Equation 2 Into Slope-Intercept Form

The second equation, \(y = 10 + 2x\), should also be written in slope-intercept form. This can be achieved by reordering the terms, resulting in \(y = 2x + 10\).

Step 3: Compare The Slopes Of The Two Equations

With both equations in slope-intercept form, it can be seen that the coefficient of \(x\) in each equation is equal, both being \(2\). Therefore, the slopes of both lines are equal.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a way to express the equation of a line. The general format is \( y = mx + b \), where:

• \( y \) represents the dependent variable (often the vertical axis on a graph).
• \( m \) is the slope of the line, which tells us how steep the line is.
• \( x \) represents the independent variable (often the horizontal axis on a graph).
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

In this structure, the slope \( m \) indicates the change in \( y \) for every one unit change in \( x \). A positive \( m \) means the line rises, while a negative \( m \) means it falls. The y-intercept \( b \) shows the starting point of the line on the graph. Understanding the slope-intercept form makes identifying parallel lines easy, as parallel lines will always have the same slope \( m \).

Equations

Equations are mathematical statements that assert the equality of two expressions. In algebra, they often involve variables, such as \( x \) and \( y \), which represent unknown numbers. Solving an equation means finding the values of the variables that make the equation true. For example, consider the equations:

• \( 2x - 12 = y \)
• \( y = 10 + 2x \)

These can both be rewritten in the slope-intercept form, as:

• \( y = 2x - 12 \)
• \( y = 2x + 10 \)

Rewriting equations in this form helps clarify their components, making it easier to compare linear equations, especially when determining if lines are parallel. Remember, parallel lines have identical slopes. Once the equations are in slope-intercept form, comparing the slopes (the coefficients of \( x \)) allows for quick determination.

Graphing

Graphing is a visual way to represent equations and inequalities. It allows us to see the relationships between variables. For linear equations, graphing involves plotting points on a coordinate plane and connecting them to form a straight line. Here's how to graph a line:

• Identify the y-intercept \( b \). Plot this point on the y-axis.
• Use the slope \( m \) to find another point. The slope \( m \) is like a set of instructions: "rise over run." If \( m = 2 \), it means for every 1 unit you move right (run), move 2 units up (rise).
• Draw a straight line through these points.

When graphing the equations \( y = 2x - 12 \) and \( y = 2x + 10 \), both will have lines with the same slope but different y-intercepts. This means they are parallel lines, as they will never meet, confirming the true nature of parallelism. Graphing helps provide a tangible visual understanding of algebraic equations.