Problem 32 - Elementary Algebra

Question

You are given one point on a line and the slope of the line. Find the coordinates of three other points on the line. $$(6,-2), m=4$$

Step-by-Step Solution

Answer

Three other points are (7, 2), (5, -6), and (8, 6).

Step 1: Understand the given information

You are given a point on the line, \((6, -2)\), and the slope of the line, \(m = 4\). We want to find three other points on the same line.

Step 2: Recall the slope formula

The slope \(m\) is defined as the change in \(y\) divided by the change in \(x\), or \(m = \frac{y_2 - y_1}{x_2 - x_1}\). We will use this to find other points by adding or subtracting from the given \(x\) and \(y\) values.

Step 3: Choose new x-coordinates

Let's choose a new \(x\)-coordinate, say \(x = 7\), which is close to \(x = 6\). Substitute into the slope formula to find the new \(y\)-coordinate.

Step 4: Calculate y-coordinates for new x-coordinate

Using the point-slope relationship \(-2 + (7-6) \times 4 = y\), substitute \(x = 7\). Calculate: \[ y = -2 + 4 = 2 \]. The new point is \((7, 2)\).

Step 5: Repeat with another x-coordinate

Now, choose \(x = 5\). Use the formula \(-2 - (6-5) \times 4 = y\). Calculate: \[ y = -2 - 4 = -6 \]. The new point is \((5, -6)\).

Step 6: Repeat with another x-coordinate

Choose \(x = 8\). Substitute into the point-slope formula: \(-2 + (8-6) \times 4 = y\). Calculate: \[ y = -2 + 8 = 6 \]. The new point is \((8, 6)\).

Step 7: Verify all points

Check all new points relate accurately to the given slope and initial point by confirming each follows the slope formula by substituting back \((6, -2)\) as \(x_1\), \(y_1\), and confirming each new point has the slope 4.

Key Concepts

Understanding Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that connects algebra and geometry through graphs and equations. It allows us to use algebraic methods to solve geometric problems. In coordinate geometry, every point on a plane is represented by an ordered pair of numbers, known as coordinates.

• The first number in the pair is the x-coordinate, which indicates the position of the point along the horizontal axis.
• The second number is the y-coordinate, which represents the position along the vertical axis.

This concept helps us visualize geometric shapes and lines on a graph, making it easier to analyze and solve related problems. For example, if we know the coordinates of a point and the slope of a line, we can find other points on that line using coordinate geometry. By understanding the relationship between points and lines on a coordinate plane, we can easily solve complex geometric problems using algebraic equations.

Exploring Slope-Intercept Form

The slope-intercept form of a linear equation is a fundamental concept in algebra and is expressed as: \( y = mx + b \). This equation describes a straight line on a graph, where:

• \( m \) is the slope of the line, representing the rate of change between y and x.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

Understanding the slope-intercept form is crucial for graphing lines and solving problems involving linear equations. The slope \( m \) indicates how steep the line is and in which direction it leans, while the y-intercept \( b \) gives us a starting point on the graph. To find a line's equation in slope-intercept form, we need either two points on the line or one point along with the slope. This form is particularly helpful when you need to quickly sketch or interpret the line on a graph.

Approaching Algebra Problem Solving

Algebra problem solving is a methodical way to tackle mathematical challenges by using algebraic equations and reasoning. Solving problems in algebra often involves a few key steps:

• Identifying what you are asked to find and collecting all given information.
• Setting up an equation using known formulas or relationships.
• Simplifying and solving the equation to find the unknown values.

For a line problem, once we have the slope and an initial point, we can use these to find other points on the line, as shown in the exercise. We do this by choosing different x-values, applying the slope to these values, and solving for the corresponding y-values. This step-by-step approach ensures clarity and accuracy in finding solutions to algebra problems, no matter their complexity.