Problem 5 - Algebra 1: Concepts and Skills

Question

find the slope and y-intercept of the equation. $$5 x-y=3$$

Step-by-Step Solution

Answer

The slope is 5 and the y-intercept is -3.

Step 1: Rearrange the Equation

Firstly, the equation \(5x - y = 3\) should be rearranged so that it is in slope-intercept form. This can be done by subtracting \(5x\) from both sides to isolate \(y\) on one side of the equation. This result in the equation: \(-y = -5x + 3\).

Step 2: Simplify the Equation

Simplify the rearranged expression to give \(y\). Multiply all terms by \(-1\) to reverse the signs. This gives the equation: \(y = 5x - 3\).

Step 3: Identify the Slope and y-Intercept

Finally, compare the equation in slope-intercept form with the standard form. The coefficient of \(x\) represents the slope (\(m\)), and the constant term represents the y-intercept (\(c\)). Here, the slope \(m = 5\) and the y-intercept \(c = -3\).

Key Concepts

Finding Slope

Understanding the slope of a line is a fundamental skill in algebra that describes how steep the line is. To find the slope from an equation in the form of ax + by = c, we first need to rearrange the equation into slope-intercept form, which is y = mx + b, where m represents the slope.

In our example, we start with the equation 5x - y = 3. To find the slope, or m, we isolate y on one side by subtracting 5x from each side to get -y = -5x + 3. Next, we multiply each term by -1 to make y positive, resulting in the final slope-intercept form y = 5x - 3. Here, the coefficient of x is 5, which is the slope. So, the slope is m = 5, which means for every one unit increase in x, y increases by 5 units.

y-intercept

The y-intercept is the point where the line crosses the y-axis. In slope-intercept form, the y-intercept is represented by b in the equation y = mx + b. It's the value of y when x is equal to zero.

In the equation y = 5x - 3, which we derived from rearranging the original equation, -3 is the constant term and thus represents the y-intercept. This means the line crosses the y-axis at (0, -3). The y-intercept is not only a point on the graph but also provides a starting value for plotting the line or understanding where the line will pass when x equals zero. So in simple terms, our y-intercept, b = -3, tells us that the starting point of the line on the y-axis is at -3.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying equations to make them more understandable or to extract specific information, such as a slope or y-intercept. It involves operations like adding, subtracting, multiplying, and dividing both sides of an equation by the same numbers or expressions to maintain equality.

In our exercise, we started with 5x - y = 3 and needed to isolate y. We subtracted 5x from both sides and then multiplied by -1 to get y by itself on one side of the equation, resulting in y = 5x - 3. Each step in this process is crucial and must be performed correctly to ensure the integrity of the equation is maintained. This example demonstrates how algebraic manipulation is used to make the equation easier to work with and to reveal the slope and y-intercept clearly.