Problem 42 - Algebra 1

Question

If \(y-x=-3\) and \(4 x+y=2,\) then \(x=?\) $$ \begin{array}{lllll} \text { (A) } 1 & \text { (B) }-1 & \text { (C) } 5 & \text { (D) }-5 \end{array} $$

Step-by-Step Solution

Answer

The solution to the set of equations given is \(x = 1\), corresponding to the option (A).

Step 1: Rearrangement of Equations

To be able to solve for \(x\), it's necessary to first rearrange our equations a bit. Let's convert both equations into terms of \(y\): From \(y-x=-3\), we get \(y=x-3\). And, from \(4x+y=2\), we get \(y=2-4x\).

Step 2: Equate Equations

Next, we equate the two derived equations from the previous step: \(x - 3 = 2 - 4x\).

Step 3: Solve for x

Adding \(4x\) to both sides of the equation gives us \(5x - 3 = 2\). Adding 3 to both sides, we then get \(5x = 5\). Dividing both sides by 5 gives us the solution: \(x = 1\)

Key Concepts

Understanding Linear Equations

Linear equations are fundamental in algebra and are the building blocks for more complex mathematical concepts. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations can always be manipulated to take the form of ax + by = c, where a, b, and c are constants, and x and y are variables. The graph of a linear equation is always a straight line.

Why is it important to understand linear equations? Because they model a plethora of real-life situations, such as calculating distances, predicting profits, or estimating costs. When solving the exercise given, we deal with two linear equations that intersect at a single point. The coordinates of this intersection point provide the solutions to x and y in the system of equations.

By effectively manipulating and solving these equations, you'll unlock the skill to tackle similar problems in subjects ranging from economics to engineering.

The Substitution Method in Practice

The substitution method is a technique used to solve systems of linear equations. This approach involves isolating one variable in one of the equations and then substituting this expression into the other equation, thereby reducing the system to a single equation with one variable.

Let's dissect the process: To isolate y in the exercise's first equation, we simply get y = x - 3. We've now expressed y in terms of x, which allows us to use this relationship as a substitute in the second equation. This eliminates y altogether, giving us a single equation in x which can be solved directly. The substitution method is powerful because it provides a clear path to finding the exact value of variables, which is exactly what you need when you're looking to find a concrete solution (such as x = ? in the given problem).

Employing this method requires understanding how to manipulate equations by adding, subtracting, multiplying, or dividing both sides - fundamental concepts in algebra that this technique reinforces.

The Art of Algebraic Problem Solving

Algebraic problem solving is not only about finding the solution to an equation; it's about understanding the methods and reasoning behind those solutions. It involves identifying strategies, executing techniques correctly, and interpreting the results.

Consider the given problem as a puzzle. We started by rearranging the equations to set a common variable, then used the substitution method to condense the system to one solvable equation. This systematic approach to solving problems is typical in algebra. It's critical to follow each step meticulously, as algebra is often about the process as much as it is about the answer.

At each step of the exercise, we applied algebraic operations to both sides of the equations to maintain equality - which is an essential principle in algebra. By adding the same value, subtracting the same value, multiplying or dividing by the same nonzero value, the equation remains balanced, and you inch closer to the solution. Mastering these operations and understanding when and how to apply them can turn a complex problem into a sequence of manageable tasks, leading you to the correct solution methodically and confidently.