Problem 52 - Understanding Elementary Algebra with Geometry

Question

Plot the points in the following set: \(\\{(x, y) | x=y+2, \text { and } x=-3,0,4\\}\)

Step-by-Step Solution

Answer

The points to plot are (-3, -5), (0, -2), and (4, 2).

Step 1: Understand the Given Conditions

We are given a set of points defined by two conditions: 1. Each point \(x, y\) must satisfy the equation \(x = y + 2\). 2. \(x\) values must be \(-3, 0, 4\).

Step 2: Substitute x = -3 into the Equation

Substitute \(x = -3\) into the equation \(x = y + 2\): \(-3 = y + 2\). Solve for \(y\): \(y = -3 - 2\) \(y = -5\).So, the point is \((-3, -5)\).\

Step 3: Substitute x = 0 into the Equation

Substitute \(x = 0\) into the equation \(x = y + 2\): \(0 = y + 2\). Solve for \(y\): \(y = 0 - 2\) \(y = -2\).So, the point is \( (0, -2)\).\

Step 4: Substitute x = 4 into the Equation

Substitute \(x = 4\) into the equation \(x = y + 2\): \(4 = y + 2\). Solve for \(y\): \(y = 4 - 2\) \(y = 2\).So, the point is \( (4, 2)\).\

Step 5: Plot the Points

Now that the points have been calculated, plot the three points on a coordinate plane: \( (-3, -5), (0, -2), (4, 2)\).\

Key Concepts

coordinate plane

The coordinate plane is a two-dimensional surface where we can plot points, lines, and curves. It's defined by two perpendicular lines called the x-axis and y-axis. These axes divide the plane into four quadrants. Each point on the plane is identified by a pair of coordinates \( (x, y) \). The first number is the x-coordinate (horizontal position), and the second number is the y-coordinate (vertical position).

When we plot points, we start at the origin \( (0, 0) \), where the x-axis and y-axis intersect. From the origin, move horizontally to match the x-coordinate and vertically to match the y-coordinate. For example, to plot \( (4, 2) \), go 4 units to the right and 2 units up.

Understanding the coordinate plane is essential for visually representing mathematical relationships and solving geometric problems.

solving equations

Solving equations is one of the fundamental skills in algebra. An equation shows that two expressions are equal, and our goal is to find the value of the variable that makes the equation true. In the given exercise, we have the equation \( x = y + 2 \), which we need to solve for different values of x.

Here are the steps for solving:

• Identify the given equation: \( x = y + 2 \).
• Substitute the given x-values (-3, 0, 4) into the equation one by one.
• Solve for the y-value that satisfies each substituted equation.

By doing this, we find the corresponding y-values for each x-value, allowing us to plot the points on the coordinate plane. This method lets us understand the relationship between x and y in the given equation.

substitution method

The substitution method is a technique used to solve systems of equations. In the context of a single equation, it involves inserting specific values into the equation to find unknown variables.

In our exercise, the steps were as follows:

• We started with the equation \( x = y + 2 \).
• We substituted \( x = -3 \) into \( -3 = y + 2 \) and solved for y, getting \( y = -5 \).
• Next, we substituted \( x = 0 \) into \( 0 = y + 2 \) and solved for y, getting \( y = -2 \).
• Finally, we substituted \( x = 4 \) into \( 4 = y + 2 \) and solved for y, getting \( y = 2 \).

By following these steps, we determined the y-values that correspond to each x-value, allowing us to plot the points accurately. The substitution method is a powerful tool for breaking down complex problems into simpler parts.