Problem 30 - Algebra 1

Question

Find the \(y\) -intercept of the graph of the equation. $$ 4 x-5 y=-35 $$

Step-by-Step Solution

Answer

The y-intercept of the graph of the given equation is 7.

Step 1: Substitute x = 0 in the equation

Firstly, we need to substitute x = 0 in the given equation, 4x - 5y = -35, which leads to -5y = -35.

Step 2: Solve for y

In the next step, we need to isolate y. This requires to divide both sides by -5, which gives us y = 7

Key Concepts

Understanding Linear Equations

Linear equations are mathematical expressions that depict a straight line when graphed on a coordinate plane. They are typically in the form of \( ax + by = c \), where \(a\), \(b\), and \(c\) are constants, and \(x\) and \(y\) are variables. The power of both \(x\) and \(y\) in a linear equation is always 1.

Linear equations have several key attributes:

• They graph as straight lines on a coordinate plane.
• Each coefficient and constant in the equation has geometric significance, such as slope or intercepts.
• The simplest form of a linear equation is \(y = mx + b\), where \(m\) represents the slope and \(b\) is the y-intercept.

Understanding these equations is crucial because they model real-world phenomena where relationships are linear, and they form the basis of more complex calculations in algebra.

How to Solve Equations for a Specific Variable

Solving equations involves manipulating a mathematical statement to isolate a variable. When dealing with linear equations like \(4x - 5y = -35\), solving for \(y\) gives the y-intercept, which is the point where the line intersects the y-axis. Here’s how we can find this step-by-step:

• Step 1: Set other variables to zero. This simplifies the equation and makes it easier to solve for the desired variable. For example, set \(x = 0\) in the equation \(4x - 5y = -35\).
• Step 2: Solve for the variable. Once substituted, the equation simplifies to \(-5y = -35\). Divide both sides by \(-5\) to isolate \(y\), leading to \(y = 7\).
• Completing these steps provides a solution in a straightforward manner, showcasing how algebraic manipulation can zero in on specific values.

By following these simple steps, you can find unknown values efficiently in many mathematical problems.

Exploring the Coordinate Plane and Its Significance

The coordinate plane is a fundamental concept in geometry and algebra, allowing us to visualize and solve problems graphically. It consists of two perpendicular axes: the horizontal (x-axis) and the vertical (y-axis).

• Points on the coordinate plane are written as ordered pairs \((x, y)\), where \(x\) is the horizontal coordinate, and \(y\) is the vertical coordinate.
• The y-intercept, specifically, represents the point where a line crosses the y-axis, meaning \(x\) is 0. This makes it an essential aspect of graphing linear equations, providing a starting point for plotting the line.

Understanding the coordinate plane enables you to better grasp the relationships portrayed by equations, and it forms a visual bridge between algebraic expressions and geometric representation. The y-intercept is an example of how information is conveyed directly through graph placement, simplifying complex ideas into manageable visuals.