Problem 38 - Algebra 1

Question

Find the value of \(y\) so that the line passing through the two points has the given slope. $$(2, y),(4,5), m=2$$

Step-by-Step Solution

Answer

The value of \(y\) in the point (2, y) so that the line passing through this point and the point (4,5) has a slope of 2 is 1.

Step 1: Identify the known points and slope

Firstly, ascertain the known quantities from the exercise. The first point is (2, y), the second point is (4, 5), and the slope \(m\) is 2.

Step 2: Insert the known quantities into the slope formula

Now insert these values into the formula \(m = \frac{{y_2 - y_1}}{{x_2 - x_1}}\). By assigning: \(x_1= 2\), \(y_1= y\), \(x_2= 4\), \(y_2= 5\), and \(m=2\), we get:\n\[2= \frac{{5 - y}}{{4 - 2}}\]

Step 3: Solve the equation for y

To solve for \(y\), first simplify the denominator to get: \[2= \frac{{5 - y}}{2}\]Then, multiply both sides of the equation by 2 to isolate \(y\): \[4= 5 - y\]Re-ordering terms, we find that: \[y= 5-4\]

Step 4: Calculate the value of y

Subtract 4 from 5, we finally get: \[y= 1\]

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This branch of mathematics allows us to solve geometrical problems by relating algebraic equations to geometrical shapes, mainly through the use of the Cartesian coordinate system.

This system uses two axes: the horizontal axis (x-axis) and the vertical axis (y-axis). Every point in the plane is defined by a pair of numerical coordinates \(x, y\), where \(x\) is the position relative to the horizontal axis, and \(y\) is the position relative to the vertical axis.

With coordinate geometry, we can easily calculate the slope of a line, find points of intersections, and solve many more geometric problems using algebraic techniques.

• Points: Defined by their coordinates (x,y).
• Lines: Often represented as linear equations.
• Slope: The measure of steepness and direction of a line.

The relationship between geometry and algebra through the use of coordinates enables precise mathematical descriptions of geometric figures.

Equation Solving

Equation solving is the process of finding the unknown values in mathematical expressions that involve variables. In our exercise, to find the value of \(y\) so that a line passing through two points has a given slope, we use the slope formula:

\[m = \frac{{y_2 - y_1}}{{x_2 - x_1}}\]

By assigning the known values from our problem:

• \(x_1 = 2\)
• \(y_1 = y\)
• \(x_2 = 4\)
• \(y_2 = 5\)
• \(m = 2\)

we substitute these into the formula to get:

\[2 = \frac{{5 - y}}{2}\]

Solving this equation involves isolating the variable \(y\). We simplify and rearrange the terms to find that \(y = 1\).

Understanding how to rearrange and solve equations is essential in mathematics, as it allows us to find values that satisfy specific conditions or constraints.

Linear Equations

Linear equations are fundamental in mathematics as they describe a straight line in a coordinate system. A typical form of a linear equation in two variables is:

\[y = mx + b\]

Where \(m\) is the slope, and \(b\) is the y-intercept. These equations graph as straight lines on the coordinate plane, and their solutions are all the points lying on the line.

The slope \(m\) indicates how much y changes for a change in x, which defines the line's steepness. The greater the absolute value of the slope, the steeper the line.

• Slope (m): Determines the angle and direction the line takes.
• Y-intercept (b): The point where the line intersects the y-axis.
• Linear relationship: A consistent rate of change between variables x and y.

Grasping the concept of linear equations is crucial as they are used across various fields in math and real-world applications, from calculating speeds to predicting future trends.