Problem 10 - Algebra 1

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understanding the equation

The given equation is \(4x - y =1\). This is a linear equation in two variables, x and y.

Step 2: Substituting the values

Now the values \(x = 1\) and \(y = 4\) from the point (1,4) are substituted into the equation. This means replacing x with 1 and y with 4 in the equation.

Step 3: Verification

The equation after substitution is \(4(1) - 4 = 1\), which simplifies to \(4 - 4 = 1\), and further simplifies to \(0 ≠ 1\). This shows that the equation does not hold true for the values \(x = 1\) and \(y = 4\).

Key Concepts

Substitution Method

The substitution method is a helpful tool for solving linear equations, especially those involving two variables. It involves replacing variables with their known values to calculate other parts of the equation. In our original exercise, we have the linear equation \(4x - y = 1\). To check if a point, say \((1, 4)\), is a solution to this equation, we use substitution.This means identifying the values of \(x\) and \(y\) in the point given. Here, \(x = 1\) and \(y = 4\). Then, substitute these values into the equation, effectively replacing any occurrence of \(x\) with 1, and \(y\) with 4. By performing these substitutions, the equation transforms into \(4(1) - 4 = 1\). This allows us to test if the equation balances, indicating if the point is indeed a solution. This straightforward approach helps simplify the problem and ascertain potential solutions effectively.

Solution Verification

After substituting the values into a linear equation, the next critical step is solution verification. This involves confirming whether the resulting equation is true or not. In our exercise, after substituting \(x = 1\) and \(y = 4\), we simplify the expression to \(4 - 4 = 1\).Breaking this down:- Calculate \(4 \times 1 = 4\). Then subtract the substituted value for \(y\), which is 4.- Simplification leads to \(4 - 4 = 0\).- Lastly, check against the original equation’s outcome, which was 1.Clearly, since \(0 eq 1\), the point \((1, 4)\) is not a solution because it does not satisfy the equation. Verification is vital in confirming whether the assumptions hold true and in ensuring that the solution aligns with the original problem setup.

Two-Variable Equations

Equations with two variables, like the linear equation \(4x - y = 1\), form a cornerstone of algebra and are crucial in various mathematical fields. These equations express a relationship between two unknowns \(x\) and \(y\). Understanding them requires recognizing that their solutions are often pairs \((x, y)\) that satisfy the equation.When dealing with two-variable equations, we're essentially looking at all the possible pairs of \(x\) and \(y\) that make the equation true. For the given example \(4x - y = 1\), potential solutions would be values of \(x\) and \(y\) which, when plugged into this equation, keep both sides equal.Learning to handle these equations involves:- Substituting different value pairs to find valid solutions.- Graphing them to visualize all possible solutions as a line.Mastering two-variable equations enhances problem-solving skills and opens the door to understanding more complex algebraic expressions.