Problem 9 - Algebra 1

Question

Tell whether the point is a solution of the equation \(4 x-y=1\). (1,0)

Step-by-Step Solution

Answer

No, the point (1,0) is not a solution for the equation \(4x - y = 1\).

Step 1: Identify the x and y values

From the point (1,0), the x-value is 1 and the y-value is 0.

Step 2: Substitute the x and y values

Insert the x and y values into the equation: \(4x - y = 1 \Rightarrow 4(1) - 0 = 1\)

Step 3: Check if both sides are equal

Simplify the equation after substitution, that results in: \(4 - 0 = 1\). After simplifying, we get that 4 equals 1, which is not true. Hence, the point is not a solution.

Key Concepts

Substitution Method

The substitution method is a foundational technique used in algebra to solve systems of linear equations. It involves replacing one variable in an equation with an expression involving the other variable. This method can simplify solving an equation or a system by reducing the number of variables you have to deal with at one time.

Let's use the substitution method in a practical example that relates closely to checking whether an ordered pair is a solution to a given linear equation. When you are given an ordered pair, like (1,0), you have specific values for 'x' and 'y'. The substitution method allows you to replace the 'x' in the equation with 1 and the 'y' with 0, resulting in a simplified expression that will reveal the truth about whether the pair is indeed a solution. As observed in the exercise, substituting the values of 'x' and 'y' leads to an equation that we can easily assess for truthfulness.

In classroom practice, teaching the substitution method requires careful guidance. Students should be encouraged to clearly identify the variables and their corresponding values and to perform substitution accurately to avoid any mix-ups that may lead to incorrect conclusions.

Ordered Pairs

Ordered pairs are a fundamental component of coordinate geometry and linear algebra. They are used to denote the location of points on a coordinate plane, with the first number representing the x-coordinate (horizontal position) and the second the y-coordinate (vertical position).

In the context of linear equations, ordered pairs are paramount as they represent potential solutions. To verify if an ordered pair is a solution, as in our exercise, you place the x-value and the y-value into the equation. If, after the substitution, the equation holds true (both sides of the equation are equal), the ordered pair is indeed a solution to the equation; if not, it isn't. The ordered pair (1,0) was tested in the given exercise, and after substitution into the equation, it did not satisfy the equation, demonstrating that it is not a solution.

Understanding ordered pairs is critical for students who are beginning to graph linear equations or evaluate points in relation to lines. It is important for educators to emphasize the order of the numbers in the pair, as mixing up x and y coordinates is a common error that can lead to confusion or incorrect results.

Linear Algebra

Linear algebra is a significant branch of mathematics focused on vectors, vector spaces, and linear transformations. It's the study of lines, planes, and their interactions, and it's instrumental in various fields like engineering, physics, computer science, and economics.

In linear algebra, equations describing lines are often written in terms of variables, such as 'x' and 'y', that can take on any values. Solving linear equations, like the one featured in this exercise, is a primary skill in linear algebra. One seeks to find the values of the variables that make the equation true, which, in geometric terms, are the points where the line (represented by the equation) crosses the axes.

The exercise demonstrates a fundamental principle in linear algebra: not every ordered pair will satisfy a given linear equation. Through examples like this, students learn that the solution to a linear equation is not just a single pair but rather a set of all possible pairs that make the equation true, forming a line when plotted. This illustrates the relationship between algebraic equations and geometric representations, a cornerstone concept of linear algebra that is crucial for students to internalize.