Problem 8 - Algebra 1: Concepts and Skills

Question

Find the \(y\) -intercept of the graph of the equation. $$ 10 x-y=-5 $$

Step-by-Step Solution

Answer

The \(y\)-intercept of the equation is \(y = 5\).

Step 1: Identify the given equation

The given equation is \(10 x - y = -5\).

Step 2: Find the y-intercept

For finding the y-intercept, set \(x\) equal to 0 and solve for \(y\). So, substitute \(x = 0\) into the equation, it gives \(10(0) - y = -5\). Simplifying this we get \(y = 5\).

Key Concepts

Linear Equations

Linear equations are fundamental in algebra and form the basis for understanding more complex mathematical concepts. A linear equation is essentially a mathematical expression representing a straight line when plotted on a graph. These equations are in the format of \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants and \(x\) and \(y\) are variables. This means the equation shows a direct relationship between \(x\) and \(y\).
When you think about linear equations, remember:

• The graph of a linear equation is always a straight line.
• The equation can be expressed in different forms: Standard Form (\(ax + by = c\)), Slope-Intercept Form (\(y = mx + b\)), or Point-Slope Form (\(y-y_1 = m(x-x_1)\)).
• The slope (\(m\)) shows the rate of change, and y-intercept (\(b\)) is where the line crosses the y-axis.

Linear equations simplify real-world problems, helping describe relationships such as distance versus time, cost versus quantity, and much more.

Solving Equations

The process of solving equations is the way to find the values of variables that satisfy the equation. In the context of linear equations, like \(10x - y = -5\), solving usually involves isolating one variable to understand its relationship with the other. Here we focus on finding specific points or intersections, such as the y-intercept.
Steps to solve a basic linear equation:

• Simplify each side of the equation if necessary (e.g., distribute, combine like terms).
• Rearrange the equation to solve for one of the variables. For y-intercepts, you'll often set \(x = 0\).
• Use basic arithmetic to isolate the variable. In the example, substituting \(x = 0\) into \(10x - y = -5\) simplifies to \(-y = -5\), giving \(y = 5\).
• Verify your solution by plugging back the value of the variable into the original equation to check if the equation holds true.

Understanding and practicing these steps helps in effectively solving any type of equation, ensuring the correct values are extracted for practical applications.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, uses algebraic equations to describe geometric principles. It allows us to represent geometric figures and solve geometric problems using a coordinate plane. This powerful tool connects algebra and geometry, enabling the visualization of algebraic equations like linear ones on a graph.
Key concepts of coordinate geometry include:

• The coordinate plane, which is divided into four quadrants by the x-axis and y-axis.
• Coordinates \((x, y)\) represent points in the plane, describing locations of points or intersections.
• The slope of a line indicates its steepness and direction, calculated as the ratio of the change in \(y\) to the change in \(x\).
• Y-intercept is another important feature, marking where a line intersects the y-axis. This is useful for making predictions about trends or experiences.

By using coordinate geometry, we can easily plot linear equations like \(10x - y = -5\) to visualize lines and understand their fundamental properties, such as slope and intercepts.