Problem 32 - Algebra 1: Concepts and Skills

Question

Find the \(y\) -intercept of the line. $$ 2 x-17 y=-51 $$

Step-by-Step Solution

Answer

The y-intercept of the line is 3.

Step 1: Identify the equation

The given equation of the line is \(2x - 17y = -51\).

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

Replacing \(x\) with \(0\) in the equation will give: \(2(0) - 17y = -51\). This simplifies to: \(-17y = -51\).

Step 3: Solve the equation for \(y\)

To solve for \(y\), divide both sides of the equation by -17. This gives: \(y = -51 / -17 = 3\).

Key Concepts

Linear Equations

Understanding linear equations is a fundamental skill in algebra and forms the basis for studying coordinate geometry. A linear equation represents a straight line on a graph and is commonly written in the form of \(y = mx + b\), where \(m\) represents the slope, and \(b\) denotes the y-intercept of the line. The slope indicates the steepness of the line, and the y-intercept is the point where the line crosses the y-axis.

In the case of our exercise, the equation \(2x - 17y = -51\) is in standard form, \(Ax + By = C\). What makes an equation \'linear\' is that the highest power of the variables \(x\) and \(y\) is 1. This ensures that the graph of the equation will be a straight line. Moreover, to quickly find the y-intercept from the standard form, you can manipulate the equation into the slope-intercept form by solving for \(y\).

Solving Equations

When it comes to solving equations, the objective is to isolate the variable of interest on one side of the equals sign to determine its value. In the focused exercise, our variable of interest is the y-coordinate of the y-intercept.

The process often involves performing inverse operations. If a variable is multiplied by a number, we divide by that number on both sides of the equation to undo the multiplication. Consistent with this principle, when we have \(2x - 17y = -51\) and we substitute \(x = 0\), the term involving \(x\) vanishes, and we are left with \( -17y = -51\). Dividing both sides by -17, a reverse operation of multiplication, we find that \(y = 3\). This value represents the y-coordinate where the line crosses the y-axis.

Solving equations correctly is essential for accurately finding the y-intercept and understanding the overall behavior of linear relationships on a graph.

Coordinate Geometry

In coordinate geometry, the concepts of lines, slopes, and intercepts come to life on a two-dimensional plane consisting of the x-axis (horizontal) and the y-axis (vertical). The intersection of these axes forms the origin, which is the center point of the coordinate plane with coordinates (0,0).

The y-intercept, the key feature of our exercise, is a specific type of intercept that lies on the y-axis. This tells us at which point a line will pass through the y-axis. For example, a y-intercept of 3 indicates that the line crosses the y-axis at the point (0,3).

The practical significance of understanding coordinate geometry is evident in various real-world applications, such as calculating trajectories, designing architectural structures, and plotting economic data. It aids students in visualizing and interpreting linear relationships and their impact within a given context.