Problem 17 - Algebra 1: Concepts and Skills

Question

Determine whether the ordered pair is a solution of the equation. $$ y=5-3 x,(2,0) $$

Step-by-Step Solution

Answer

The ordered pair (2,0) is not a solution to the equation \(y = 5 - 3x\).

Step 1: Substitution

Substitute the values from the ordered pair (2, 0) into the equation \(y = 5 - 3x\). Here, x is 2 and y is 0,thus the equation becomes \(0 = 5 - 3 * 2\).

Step 2: Calculate

Calculate the result on the right side of the equation: \(5 - 3 * 2 = 5 - 6 = -1\). This yields the equation \(0 = -1\).

Step 3: Compare

Now compare the results on both sides of the equation. We have: \(0 = -1\). As we see, the left and right sides of the equation are not equal, meaning this ordered pair does not satisfy the equation.

Key Concepts

Substitution Method

The substitution method is a powerful technique used primarily to determine if an ordered pair is a solution to a given equation. This involves replacing the variables in the equation with the respective values from the ordered pair. By doing this, you're effectively "testing" whether the numbers fit logically into the equation.Let's consider how this works with the example equation:- You start with the equation, which in this case is \( y = 5 - 3x \).
- From the ordered pair, identify your \( x \) and \( y \) values. Here, \( (2, 0) \) means: \( x = 2 \) and \( y = 0 \).
- Substitute these values into the equation, replacing \( y \) with 0 and \( x \) with 2. This transforms the equation into \( 0 = 5 - 3 \times 2 \).The goal of substitution is to see if the equation holds true with the given numbers. If both sides of the equation are equal after substitution, then the ordered pair is indeed a solution.

Ordered Pairs

Ordered pairs are an essential concept in mathematics, especially when dealing with coordinates, functions, and equations. An ordered pair typically consists of two elements written in a specific order within parentheses, like this: \( (x, y) \).In the context of solving equations, the ordered pair represents a potential solution:- The first element, \( x \), is the input or the independent variable.
- The second element, \( y \), is the output or the dependent variable.
These pairs are used to map points on a coordinate plane, and when substituted into an equation, can verify a solution. In our example of the ordered pair \( (2, 0) \): - \( x = 2 \) is substituted into the equation for \( x \).- \( y = 0 \) is substituted into the equation for \( y \). Effectively, these serve as potential solutions that we check against the given equation to find whether or not they satisfy it.

Linear Equations

Linear equations are a fundamental type of equation in algebra that graph as straight lines. They are typically written in the form \( y = mx + b \), where:- \( y \) and \( x \) are variables.
- \( m \) is the slope, which indicates the steepness or tilt of the line.
- \( b \) is the y-intercept, representing the point where the line crosses the y-axis.These equations are used in various applications, from geometry to physics, due to their straightforward relationships between variables. In the context of our original problem:- The equation \( y = 5 - 3x \) is a linear equation.- The task was to determine if the ordered pair \( (2, 0) \) satisfied it by substituting the values and checking the equality.Once you substitute the values and find both sides unequal, as in our example where \( 0 eq -1 \), it indicates that this particular ordered pair does not lie on the line represented by the equation.