Problem 14 - Elementary Algebra

Question

\(y=-2 x+3 ;(1,1),(1.5,0),(3,-1),(0,3),(-1,5)\)

Step-by-Step Solution

Answer

Points (1, 1), (1.5, 0), (0, 3), and (-1, 5) are on the line.

Step 1: Understanding the Exercise

We are given a linear equation: \( y = -2x + 3 \). We need to determine if the given points lie on the line defined by this equation. The points are: \((1, 1), (1.5, 0), (3, -1), (0, 3), (-1, 5)\).

Step 2: Testing the First Point (1,1)

Insert \(x = 1\) into the equation to find \(y\):\[ y = -2(1) + 3 = 1 \]Since \(y = 1\) matches the given point (1, 1), this point is on the line.

Step 3: Testing the Second Point (1.5, 0)

Insert \(x = 1.5\) into the equation to find \(y\):\[ y = -2(1.5) + 3 = 0 \]Since \(y = 0\) matches the given point (1.5, 0), this point is on the line as well.

Step 4: Testing the Third Point (3,-1)

Insert \(x = 3\) into the equation to test for \(y\):\[ y = -2(3) + 3 = -3 \]Since \(y = -3\) does not match \(-1\) in the point (3, -1), this point is not on the line.

Step 5: Testing the Fourth Point (0,3)

Insert \(x = 0\) into the equation to test for \(y\):\[ y = -2(0) + 3 = 3 \]Since \(y = 3\) matches the given point (0, 3), this point is on the line.

Step 6: Testing the Fifth Point (-1,5)

Insert \(x = -1\) into the equation to test for \(y\):\[ y = -2(-1) + 3 = 5 \]Since \(y = 5\) matches the given point (-1, 5), this point is on the line.

Key Concepts

Graphing Points

When working with linear equations, understanding how to graph points is crucial. Each point on a graph is defined by a pair of numbers in parentheses, such as

• (1,1)
• (1.5,0)
• (3,-1)
• (0,3)
• (-1,5)

These are known as ordered pairs. The first number represents the x-coordinate (horizontal position) and the second number is the y-coordinate (vertical position).
To graph these points:

• Start at the origin, which is (0,0)
• Move right for positive x-values, left for negative x-values
• Then move up for positive y-values, down for negative y-values

By plotting these points on a graph, you can visualize where they lie in relation to each other and see if they form a straight line, as described by the linear equation.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, allows us to represent geometric shapes as algebraic equations. With this method, you can see that each point on a graph corresponds to a unique set of coordinates

• (x, y)
• Use these coordinates to test whether a point lies on a particular line defined by a linear equation

In our exercise, we used the equation \(y = -2x + 3\).
We insert different x-values from the given points into this equation to see if the calculated y-value matches. If it does, the point is on the line. In this way, coordinate geometry provides a bridge between numeric and geometric understanding of lines.
This bridging is especially useful when trying to understand graph behavior and positioning without needing to rely solely on physical graph plotting. It brings precision and efficiency into working with geometry in an algebraic form.

Slope-Intercept Form

The slope-intercept form is a way to express linear equations of the form \(y = mx + b\). Here, \(m\) represents the slope, and \(b\) is the y-intercept of the line.
In our equation \(y = -2x + 3\), the slope \(m\) is -2, indicating that for every unit increase in \(x\), \(y\) decreases by 2. The y-intercept \(b\), which is 3, tells us that the line crosses the y-axis at the point (0,3).
The slope is a measure of the steepness or incline of the line. A positive slope rises, and a negative slope falls as we move along the x-axis.
Meanwhile, the y-intercept is simply where the line touches the y-axis. Understanding both these components helps in graphing a line quickly and correctly. To apply this:

• Start by plotting the y-intercept on the graph
• From there, use the slope to determine the next points

This method makes it easy to visualize and draw the line on a coordinate plane, providing a simple visual linkage between algebra and geometry.