Problem 78 - Algebra 1

Question

Use the equation \(2 x+7 y=14\). What is the \(y\) -intercept?

Step-by-Step Solution

Answer

The y-intercept of the equation \(2x + 7y = 14\) is 2.

Step 1: Understand the problem

The equation given is \(2x + 7y = 14\). The goal is to find the y-intercept, which is the point where the line crosses the y-axis. This occurs when \(x = 0\).

Step 2: Substitute \(x = 0\) into the equation

Substitute \(x = 0\) into the equation \(2x + 7y = 14\) to find \(y\). The equation becomes \(2(0) + 7y = 14\) which simplifies to \(7y = 14\).

Step 3: Solve for \(y\)

Solve the equation \(7y = 14\) for \(y\) by dividing both sides by 7. The solution is \(y = 14 / 7 = 2\). Therefore, the y-intercept is 2.

Key Concepts

Linear Equation

A linear equation is an algebraic expression that describes a straight line when graphed on a coordinate plane.
These equations are typically represented in the form \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) and \( y \) are variables.
Linear equations can be rearranged into different forms, such as the slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

• In the given equation, \( 2x + 7y = 14 \), the coefficients of \( x \) and \( y \) represent the slope (when rearranged) and impact the position and angle of the line on the graph.
• A linear equation will graph as a straight line across a coordinate plane, making it easier to predict and understand the relationship between \( x \) and \( y \) values.

Understanding linear equations is fundamental when analyzing relationships and making predictions based on data.

Solving Equations

Solving equations involves finding the values of variables that satisfy the equation.
For a linear equation like \( 2x + 7y = 14 \), we often solve for \( y \) to find specific points, particularly intercepts, on the graph.
Here's how you find the solution:

• First, identify what you're solving for—in this example, it’s the y-intercept.
• Set \( x = 0 \) because the y-intercept is where the line crosses the y-axis.
• Substitute \( x = 0 \) into the equation, which simplifies it to \( 7y = 14 \).
• To isolate \( y \), divide both sides by 7, resulting in \( y = 2 \).

These steps demonstrate the methodical approach essential for solving equations effectively.
Equations can be tackled using various techniques like substitution, elimination, or graphing, depending on the complexity and type of equation at hand.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, connects algebra and geometry using a system called the coordinate plane.
It involves plotting points, lines, and shapes using coordinates (\( x, y \)) and exploring their properties.
In the case of finding the y-intercept:

• The y-intercept is the point where a graph crosses the y-axis. This point has coordinates \( (0, y) \).
• In the equation \( 2x + 7y = 14 \), the y-axis crossing point (y-intercept) was found when substituting \( x = 0 \).
• This converts the algebraic solution into a spatial understanding, helping visualize how the equation translates into a line that intersects the y-axis at (0, 2).

Coordinate geometry allows us to graphically represent and analyze linear equations, helping to interpret real-world phenomena and solve geometry problems efficiently.