Problem 5 - Intermediate Algebra

Question

$$ y=-x+3 \quad \begin{array}{l|llll} x & -2 & -1 & 0 & 4 \\ \hline y & & & & \end{array} $$

Step-by-Step Solution

Answer

For the given \( x \) values, the corresponding \( y \) values are 5, 4, 3, and -1.

Step 1: Understand the Equation

The equation given is a linear equation in the form \( y = -x + 3 \). This represents a straight line, where for every value of \( x \), you can find the corresponding value of \( y \).

Step 2: Substitute the First Value of x

Substitute \( x = -2 \) into the equation to find \( y \): \[y = -(-2) + 3 = 2 + 3 = 5\] So, when \( x = -2 \), \( y = 5 \).

Step 3: Substitute the Second Value of x

Substitute \( x = -1 \) into the equation to find \( y \): \[y = -(-1) + 3 = 1 + 3 = 4\] So, when \( x = -1 \), \( y = 4 \).

Step 4: Substitute the Third Value of x

Substitute \( x = 0 \) into the equation to find \( y \): \[y = -0 + 3 = 3\] So, when \( x = 0 \), \( y = 3 \).

Step 5: Substitute the Fourth Value of x

Substitute \( x = 4 \) into the equation to find \( y \): \[y = -(4) + 3 = -4 + 3 = -1\] So, when \( x = 4 \), \( y = -1 \).

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra. A linear equation is one where the variables appear only to the first power and are not multiplied by each other. Consider the linear equation given in the exercise: \( y = -x + 3 \). This means no matter what x value you plug into the equation, the y value you get will plot a straight line when graphed.
Solving the equation involves finding the y value that corresponds to a given x value. Here is how you do it:

• Start by taking the given equation.
• Substitute the x value into the equation.
• Perform the arithmetic operations to solve for y.

For instance, substituting \( x = -2 \), you perform the operations \(-(-2) + 3\), which results in \( y = 5 \). This way, you determine that when \( x = -2 \), \( y = 5 \). Repeat for any other values of x provided.

Coordinate System

In mathematics, a coordinate system is used to uniquely determine the position of an element within space, most commonly expressed in 2D as the x and y axes. These axes intersect at a point called the origin, typically marked as (0,0).
Understanding how to plot points onto a coordinate system based on a linear equation model, like \( y = -x + 3 \), helps visualize solutions. Each ordered pair (x, y) found from substituting x values into the equation can be plotted on this plane.
The advantage of using a coordinate system is clear:

• It helps in visualizing the relationship between values.
• Once points are plotted, you can see the line they form.
• This line represents all possible solutions of the equation.

For example, plotting points such as (-2, 5), (-1, 4), (0, 3), and (4, -1) onto the coordinate grid, outlines the line described by the equation. This not only represents solutions graphically but also aids in understanding the problem scenario visually.

Substitution Method

The substitution method is a technique used to find solutions of equations, particularly useful in solving systems of linear equations, but also applicable here. The principle of substitution involves replacing a variable with its known value.
Using the equation \( y = -x + 3 \), we apply substitution by putting specific x values into the equation one at a time:

• Take each x value provided, like -2 or 4.
• Replace x in the equation with this value, forming a simple arithmetic expression.
• Solve for y to find the corresponding point on the graph.

This method ensures that every calculation is straightforward and accurate. For example, after substituting \( x = 0 \), you calculate \( y = -0 + 3 \), giving \( y = 3 \). By repeating these steps for each x value, the mystery of how y is determined from x becomes clear, showing concrete, precise results such as (0, 3) or (4, -1). The substitution process breaks down the equation into manageable steps, making it easier to understand the relationship between the two variables. This technique aids significantly when dealing with more complex systems later on.