Problem 8 - Algebra 1

Question

Tell whether the point is a solution of the equation \(4 x-y=1\). (1,3)

Step-by-Step Solution

Answer

Yes, the point (1,3) is a solution to the equation \(4 x-y=1\).

Step 1: Identify the coordinates in the point

The given point is (1,3). This means that x = 1 and y = 3.

Step 2: Substitute the values into the equation

Plug the values of x and y into the given equation in their respective places. So we'll have \(4(1) - 3 = 1\).

Step 3: Solve the equation

If the left-hand side equals the right-hand side after substituting, then the point is a solution to the equation. We'll have \(4 - 3 = 1\). After simplifying, we get 1 = 1.

Step 4: Check if the equation is true

After calculating and simplifying, our equation, 1 = 1, is a true statement which means our given coordinate is indeed a solution to the equation.

Key Concepts

Coordinate Substitution

In algebra, coordinate substitution is a handy method to verify if a point satisfies a given equation. It starts by identifying the coordinates of the point, usually denoted as \(x, y\). For the point \(1,3\), this means \(x = 1\) and \(y = 3\). To use coordinate substitution effectively, follow these steps:

• Identify the values for \(x\) and \(y\) from the point you are given.
• Substitute these values into the equation in place of \(x\) and \(y\).
• In our example equation \4x - y = 1\, replacing \(x\) with 1 and \(y\) with 3 gives the expression \[4(1) - 3\].

This substitution helps us transform the abstract equation into a concrete expression that we can solve to determine if the point lies on the line represented by the equation.

Solution Verification

Solution verification is a crucial step to ensure that the coordinates indeed satisfy the equation. It involves solving the substituted equation and checking if both sides are equal. After substitution, we solve the expression \[4(1) - 3\] and verify if it results in the right-hand side of the original equation which is 1.

• Simplify the equation. For our example, simplifying \4 - 3\ yields 1.
• Compare this result to the right-hand side of the original equation.
• If both sides match, the point is a true solution.

The process confirms that the coordinate (1,3) is indeed a solution to the equation, as the calculated result is 1, which matches the required value.

Linear Equations

Linear equations are mathematical expressions that graph as straight lines on a coordinate plane. They typically take the form \ax + by = c\, where \(a\), \(b\), and \(c\) are constants, and \(x\) and \(y\) are variables. For example, in the equation \4x - y = 1\, the \(a\) and \(b\) values are 4 and -1, respectively, and \(c\) is 1.
To further understand linear equations, consider:

• They can have infinitely many solutions, forming a line.
• Each solution represents a point where the line crosses the coordinate plane.
• Checking points is just an algebraic way to see if they lie on this line.

Understanding linear equations and their properties is essential in analyzing and solving algebraic problems. The verification process using coordinate substitution reinforces these fundamental concepts.