Problem 6 - Algebra 1: Concepts and Skills

Question

Find the \(y\) -intercept of the graph of the equation. $$ 6 x+3 y=51 $$

Step-by-Step Solution

Answer

The y-intercept of the given equation is \(y=17\).

Step 1: Identify the equation

The equation given is \(6x+3y=51\). This is a linear equation in two variables.

Step 2: Set x to zero

We need to find out the point where the line represented by the equation crosses the y-axis. This happens when x = 0. So, plug x = 0 in the equation.

Step 3: Solve for y

Substitute x = 0 in the equation \(6x+3y=51\). The equation becomes \(3y=51\). Solve for y by dividing each side of the equation by 3. After dividing, we get \(y=17\).

Key Concepts

Understanding Linear Equations

When we encounter an expression such as 6x + 3y = 51, we're looking at a linear equation. In its simplest form, a linear equation describes a straight line when plotted on a graph. These equations typically have one or more variables (in this case, x and y) and are characterized by each term being either a constant or the product of a constant and a single variable.

One of the fundamental aspects of linear equations is their slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept. The y-intercept is a crucial piece of information because it tells you where the line crosses the y-axis. By finding the y-intercept, we essentially determine the value of y when x is zero. This is the starting point in graphing a linear equation and helps in understanding the placement of the line on a standard coordinate grid.

Graphing Linear Equations

To visualize what the equation 6x + 3y = 51 represents, we graph it.
Graphing linear equations involves plotting points on a grid and connecting them to form a straight line. A smart starting point is identifying key components, such as the y-intercept, which we've determined is (0, 17) in this equation.

Step 1: Start by plotting the y-intercept on the graph. This is where the line will cross the y-axis.
Step 2: Determine another point. A simple way is to alter the value of one variable and solve for the other. If we let x be 1, for example, we can solve the equation for y and plot this second point.
Step 3: Draw a straight line through the points. The straightness of the line is crucial as it confirms the equation's 'linear' qualifier. Any linear equation will graph as a straight line, and every point on that line will satisfy the original equation.

Algebraic Problem-Solving

Algebraic problem-solving is a methodical approach to solving equations and understanding their implications. It involves a series of steps to simplify and solve equations for unknown variables. Here's how we can apply this method to our example:

Step 1: Identify the structure of the equation. With our given equation, we see two variables and constants that indicate it's a linear equation.
Step 2: Look for what the question is asking. If it's the y-intercept, we know we need the value of y when x equals zero.
Step 3: Manipulate the equation accordingly, which in this case means substituting x with zero and solving for y.
Step 4: Perform any necessary operations to isolate the variable in question, as we did by dividing each side of 3y = 51 by 3, yielding y = 17. In algebraic problem-solving, it is crucial to be systematic, work step-by-step, and double-check each calculation to ensure accuracy and understanding of the problem's resolution.