Problem 39 - Algebra 1

Question

Find the value of \(y\) so that the line passing through the two points has the given slope. $$(0,-2),(2, y), m=3$$

Step-by-Step Solution

Answer

The value of 'y' is 4.

Step 1: Insert Known Values into the Slope Formula

To start, let's write down the formula for slope and insert the given values into it: The provided points are (0,-2) as \((x_1, y_1)\) and (2, y) as \((x_2, y_2)\). So the formula changes to: \(m = \frac{(y - (-2))}{(2 - 0)}\)

Step 2: Simplify the Formula

Simplify the above formula. This will turn into: \(3 = \frac{(y + 2)}{2}\), since given m = 3.

Step 3: Solve for y

Multiply both sides by 2 to solve for 'y' as a final answer and then subtract 2 from both sides: \(y = 3*2 - 2\).

Key Concepts

Point-Slope Form

Understanding the point-slope form is crucial for anyone learning algebra, especially when it comes to graphing linear equations. It is written as \( y - y_1 = m(x - x_1) \).
This equation expresses the relationship of a linear equation passing through a specific point \( (x_1, y_1) \) with a slope of 'm'. In our exercise, the goal was to find the value of 'y' for a line through the points \( (0,-2) \) and \( (2, y) \), with a given slope of 3. By identifying the known x and y values, and the slope, we can easily arrive at the point-slope form of the line.
In practice, you can rearrange the point-slope form to isolate 'y' on one side which makes it resemble the more familiar slope-intercept form, \( y = mx + b \).
This versatility in forms helps with various algebraic manipulations, making point-slope form a powerful tool in the algebra toolkit.

Linear Equations

Linear equations form the basis of a wide range of real-world problems and are characterized by their straight-line graphs. The general form of a linear equation in two variables, x and y, is \( Ax + By + C = 0 \),
where A, B, and C are constants. The slope-intercept form \( y = mx + b \)
is another popular way to represent a linear equation, showing how y changes with x, multiplied by the slope 'm', plus the y-intercept 'b'.
In our exercise, we use the concept of slope, defined as 'rise over run' or the change in y over the change in x, to determine the missing coordinate. Understanding how to graph these equations and analyze the slope allows for a better comprehension of how variables are connected linearly.

Algebraic Manipulation

Algebraic manipulation is the process of transforming equations or expressions to a desired form using algebraic rules. This process is essential to solving equations and simplifying expressions. During the exercise, we executed a series of algebraic manipulations to find the unknown value of 'y'.
Initially, we inserted the known values into the slope formula. After simplifying, we used multiplication to eliminate the fraction, followed by subtraction to isolate 'y'.
Such skills are fundamental in all fields of mathematics, as they apply to a broad spectrum of problems. Strengthening your ability in algebraic manipulation will make it easier to handle more complex equations and lead you successfully through the labyrinth of algebra.