Problem 42 - Algebra 1

Question

Find the value of \(y\) so that the line passing through the two points has the given slope. $$(-1,5),(3, y), m=5$$

Step-by-Step Solution

Answer

The value of y that makes the line passing through the points (-1,5) and (3,y) have a slope of 5 is y = 25.

Step 1: Write out the Slope Formula

Identify the coordinates of the two points and the given slope. Plug these values into the slope formula. The slope \(m\) is given by the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\), where \((x_1, y_1) = (-1,5)\) and \((x_2, y_2) = (3, y)\), and \(m = 5\) is the given slope.

Step 2: Substitute the Given Values into the Equation

Substitute the given values into the equation: \(5 = \frac{y - 5}{3 - (-1)}\).

Step 3: Simplify and Solve for y

Simplify this equation to solve for \(y\). Start by simplifying the denominator: \(5 = \frac{y - 5}{4}\). Then, multiply both sides by 4 to solve for \(y\): \(20 = y - 5\). Finally, add 5 to both sides to find the value of \(y\): \(y = 25\).

Key Concepts

Understanding Slope

The concept of slope is a fundamental aspect of algebra and coordinate geometry. Simply put, the slope of a line measures how steep it is. It tells us how much the line goes up or down as we move along it. The formula for the slope, denoted by \( m \), is given by \( m = \frac{y_2 - y_1}{x_2 - x_1} \). This formula involves two points: \((x_1, y_1)\) and \((x_2, y_2)\), which are coordinates on a plane.

• The numerator \( (y_2 - y_1) \) indicates the vertical change — how much the line goes up or down.
• The denominator \( (x_2 - x_1) \) indicates the horizontal change — how much the line moves left or right.

A positive slope means the line inclines upwards, while a negative slope indicates it declines. A zero slope signifies a horizontal line, and an undefined slope is associated with vertical lines because the denominator becomes zero (division by zero). Understanding these aspects helps in analyzing and graphing linear relationships effectively.

Exploring Linear Equations

Linear equations are equations of the first degree, meaning they involve no exponents higher than one. They form straight lines when graphed on a coordinate plane. The general form of a linear equation is \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept. The y-intercept is the point where the line crosses the y-axis.

When working with linear equations, you often have to determine one of these components given the other values. For example, if the slope and a point on the line are known, the equation can be solved to find another point or the y-intercept.

• Linear equations can model real-world situations, such as predicting outcomes in data trends and financial forecasts.
• It’s crucial to understand the behavior of these equations since they form the basis for more complex algebraic concepts.

Linear equations are simple yet powerful tools for representing constant rates of change within a system.

Basics of Coordinate Geometry

Coordinate geometry, also known as analytic geometry, uses a coordinate system to graph and study the positions of points. It blends algebra and geometry, allowing for the graphical representation of equations and the algebraic representation of geometric principles.

In this space, each point is specified by a pair of numerical coordinates \((x, y)\). These coordinates are ordered pairs that measure distances along perpendicular lines called the axes.

• The x-axis runs horizontally, with points measured left and right from the origin.
• The y-axis runs vertically, with points measured up and down from the origin.
• Each point \((x, y)\) has a precise location on the plane.

Using coordinate geometry, you can solve a variety of geometric problems, including those that involve finding the gradient of lines, the distance between points, or identifying whether lines are parallel or perpendicular. Understanding this subject is key to exploring relationships between different algebraic equations and their geometric counterparts.