Problem 29 - Algebra 1: Concepts and Skills

Question

Find the \(y\) -intercept of the line. $$ y=13 x+26 $$

Step-by-Step Solution

Answer

The y-intercept of the line defined by the equation \(y = 13x +26\) is \(26\).

Step 1: Find the y-intercept

The y-intercept is the point where the graph of a function or equation crosses or 'intercepts' the y-axis. As \(y=mx+b\), where \(m\) is the slope and \(b\) is the y-intercept, from the provided equation \(y=13x+26\), it is clear that the y-intercept \(b=26\).

Step 2: Write the equation in standard form

Rearrange and simplify the equation.

Step 3: Apply the solution method

Use factoring, quadratic formula, substitution, or other methods.

Step 4: Verify the solution(s)

Check solutions in the original equation.

Step 5: State the final answer

List all valid solutions.

Step 6: Conclude with the answer

The y-intercept of the line defined by the equation \(y = 13x +26\) is \(26\).

Key Concepts

linear equations

Understanding linear equations is crucial for solving a variety of problems in algebra. At its core, a linear equation represents a straight line when plotted on a two-dimensional graph. It is called 'linear' because it signifies a constant rate of change, which can be visualized as a line's slope. The general form of a linear equation is \(y = mx + b\), where \(m\) stands for the slope or the steepness of the line, and \(b\) represents the y-intercept, which is where the line crosses the y-axis.

When you're given a linear equation like \(y = 13x + 26\), you can identify two primary characteristics: the slope (13) indicates how steeply the line rises or falls as you move along the x-axis, and the y-intercept (26) tells you the exact point on the y-axis where the line would pass through. It is vital to grasp these concepts to analyze the behavior of the line associated with such equations effectively.

slope-intercept form

The slope-intercept form of a linear equation is a way to easily convey the information needed to graph a line. This form is written as \(y = mx + b\), where \(m\) represents the slope, and \(b\) is the y-intercept. Intuitively, the slope is a measure of how tilted the line is; a larger slope means a steeper line. Positive slopes tilt upwards, whereas negative slopes tilt downwards.

In the equation \(y = 13x + 26\), using the slope-intercept form, you can immediately see that 13 is the slope and 26 is the y-intercept. This presentation makes it straightforward to sketch the line on a graph by starting at the y-intercept and using the slope to find another point on the line. For those new to the concept, remembering 'rise over run' helps in understanding the slope: it is the ratio of the vertical change (rise) to the horizontal change (run) as you move from one point to another along the line.

graphing lines

Graphing lines is a fundamental skill in algebra that involves taking an equation and translating it into a visual representation on a coordinate plane. To graph a line from an equation in slope-intercept form, such as \(y = 13x + 26\), you would typically start by plotting the y-intercept (0,26) on the y-axis. This point is where the line crosses the y-axis.

From there, you would use the slope to find another point. Since the slope is 13, or \(\frac{13}{1}\), you would move up 13 units vertically (the 'rise') and go 1 unit to the right horizontally (the 'run'). Placing a second point at this new location and drawing a line through both points will give you the graph of the equation. The y-intercept and the slope are the two key pieces of information that make graphing a straightforward process. To ensure you're graphing lines correctly, always double-check that your y-intercept is accurate and use the slope to guide you from that point.