Problem 31 - Algebra 1: Concepts and Skills

Question

Find the \(y\) -intercept of the line. $$ 3 x-4 y=16 $$

Step-by-Step Solution

Answer

The y-intercept of the line is -4.

Step 1: Identify the equation's form

In order to find the intercept, one needs to recognize the form of the equation. The given equation, \(3x - 4y = 16\), is in standard form, \(Ax + By = C\), where A, B and C represent integer coefficients.

Step 2: Set x to zero

Since the y-intercept is asked, x is substituted with 0 in the equation to solve for the value of y, this results in \(-4y = 16\)

Step 3: Solving for y

Solving \(-4y = 16\) for y, it is found by dividing both sides of the equation by -4, hence y = -4

Key Concepts

Standard Form Equation

Understanding the standard form of a linear equation is crucial in algebra. It's typically written as \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers, and \(A\) or \(B\) can't both be zero. In this format, it's easy to identify the slope and y-intercept which are key properties of a line.

In the given exercise, the equation \(3x - 4y = 16\) represents a line in standard form. To find the y-intercept, one should remember that this is the point where the line crosses the y-axis. At this point, the value of \(x\) is always zero because the y-axis is the line where all points have an \(x\) coordinate of zero. Therefore, setting \(x\) to zero will isolate \(y\) and reveal the y-intercept.

Solving Linear Equations

Linear equations are the foundation of algebra and represent lines on a graph. Solving these requires algebraic manipulation to get the variable you're solving for by itself on one side of the equation. The main goal is to isolate the variable, often achieved by performing operations such as addition, subtraction, multiplication, and division.

In our case, the objective is to solve for \(y\) to find the y-intercept. Starting with the standard form equation \(3x - 4y = 16\), and knowing that the y-intercept occurs when \(x = 0\), you replace \(x\) with zero, reducing the equation to \(-4y = 16\). The final step is to divide both sides by \(-4\) to isolate \(y\), giving the y-intercept as \(y = -4\). It is crucial that each algebraic operation performed on one side of the equation is also done to the other side to maintain equality.

Algebraic Methods

Algebraic methods encompass various techniques used to manipulate and solve equations. These methods are fundamental in simplifying expressions, solving for unknowns, and understanding the relationships between variables. Common methods include using properties of equality, factoring, and distributing.

The exercise provided is a perfect example of applying an algebraic method to solve for the y-intercept of a line. After setting \(x\) to zero based on our understanding of the y-intercept on a graph, we simplify the equation to isolate \(y\). This involves dividing each term by the coefficient of \(y\), which is \(-4\) in this case. This division is an algebraic method that ensures we solve for \(y\) accurately. The result is a clear, singular value for \(y\), concluding the method.