Problem 7 - Algebra 1: Concepts and Skills

Question

Find the \(y\) -intercept of the graph of the equation. $$ -2 x-8 y=16 $$

Step-by-Step Solution

Answer

\(-2\)

Step 1: Setting x to zero

First, the x value is set to zero, because the y-intercept is the point where the line crosses the y-axis, and at this point, the x-coordinate is zero. This gives the equation -2(0) - 8y = 16, which simplifies to -8y = 16.

Step 2: Solve for y

The next step is to solve for y, by dividing both sides of the equation by -8. This gives y = 16 / -8.

Step 3: Simplify the equation

Simplify the equation to get the final solution. This gives y = -2, which is the y-intercept of the graph of the equation.

Key Concepts

Understanding Linear Equations

When dealing with linear equations, you are looking at mathematical expressions that, when graphed, will produce straight lines. A linear equation typically appears in the form of \( Ax + By = C \), where \( A \), \( B \), and \( C \) are constants. In the case of the exercise provided, we have the equation \( -2x - 8y = 16 \).

This equation represents a straight line and finding its features, like the y-intercept, is key to understanding its graph. The y-intercept itself is a point where the line crosses the y-axis, which, by definition, happens when \( x = 0 \). In a way, the y-intercept tells us where our line starts or what value of \( y \) we have when \( x \) is nothing.

Graphing Lines from Equations

Graphing a line requires an understanding of its slope and intercepts. To graph the line represented by the equation \( -2x - 8y = 16 \), you could rewrite it in slope-intercept form - which is \( y = mx + b \), where \( m \) represents the slope and \( b \) the y-intercept - or find specific points like intercepts and plot them.

To graph without the slope-intercept form, find both the x-intercept (where the line crosses the x-axis) and y-intercept (where it crosses the y-axis). You've already found that the y-intercept for this equation is \( y = -2 \) by setting \( x \) to zero. This gives you a starting point on the y-axis at \( (0, -2) \). You might also set \( y \) to zero to find the x-intercept for a complete graph.

Solving for y

In order to find the y-intercept directly from an equation, you can 'solve for y'. This method rearranges the equation so that y is by itself on one side of the equals sign. For the provided exercise, solving for y is demonstrated through these steps. First, you neutralize the x component by setting it to zero, which is logical since you're interested in the point where the line meets the y-axis (and at this point, the value of x is zero).

Once x is out of the equation, you're left with a simple algebraic expression that can be easily solved for y: \(-8y = 16\). By dividing both sides by the coefficient in front of y, in this case, -8, you isolate y and find that it is -2. Thus, the y-intercept, the point where the line intersects with the y-axis, is simply the point (0, -2). Remember, the y-intercept is a specific kind of solution where we're solely interested in the vertical (y) component when our horizontal (x) component is non-existent.