Problem 77 - Algebra 1: Concepts and Skills

Question

Use the substitution method to solve the linear system. $$ \begin{aligned} &x-y=0\\\ &3 x+4 y=14 \end{aligned} $$

Step-by-Step Solution

Answer

The solution is \(x = 2\) and \(y = 2\), or as an ordered pair, (2, 2).

Step 1: Isolate One Variable

Isolate any of the variables. Because \(x\) is already by itself on one side of the equation 1, \(x = y\).

Step 2: Substitute the Isolated Variable into the Second Equation

Substitute \(x\) (which we found to be equal to \(y\)) into equation 2: \(3y + 4y = 14\).

Step 3: Solve for the Substituted Variable

Solving the equation from step 2 gives us \(7y = 14\). Dividing both sides by 7, we find that \(y = 2\).

Step 4: Solve for the Second Variable

Substitute \(y = 2\) back into equation 1 \(x = y\). This yields \(x = 2\).

Step 5: State the Final Answer

Our final solution is \(x = 2\) and \(y = 2\), or as an ordered pair, (2, 2).

Key Concepts

Solving Linear Equations

The process of finding the value of variables that satisfy a linear equation is known as solving linear equations. Linear equations are algebraic expressions where each term is either a constant or the product of a constant and a single variable.

For instance, in our exercise, we have the equations \(x-y=0\) and \(3x+4y=14\). To solve this system, we seek values for \(x\) and \(y\) that make both equations true simultaneously. The substitution method simplifies this by reducing the two-variable system to a single variable equation, which can then be solved using basic algebraic principles.

Algebraic Techniques

Algebraic techniques encompass a variety of methods used to manipulate and solve equations. These can include adding, subtracting, multiplying, or dividing both sides of an equation by the same number (to maintain equality), as well as more complex operations like factoring, expanding, or simplifying expressions.

In our exercise, after isolating \(x\) in the first equation, we substituted it in the second equation, transforming it into \(3y+4y=14\). Combining like terms resulted in \(7y=14\), a simpler equation that we could solve by dividing both sides by 7 to isolate the variable \(y\).

Isolating Variables

Isolating variables is an important skill in algebra that involves rearranging an equation to get a single variable on one side of the equal sign and everything else on the other side. This technique makes it easier to solve for that variable.

For example, in the first step of our solution, we noticed that \(x\) was already isolated in the first equation: \(x = y\). This presented an opportunity to use substitution directly without further manipulation, clearly illustrating how isolating variables simplifies the problem-solving process.

Systems of Linear Equations

A system of linear equations consists of two or more linear equations involving the same set of variables. The goal is to find values for the variables that satisfy all equations in the system simultaneously. These systems can be solved using various methods, including graphing, substitution, elimination, and matrix methods.

In the substitution method, as seen in our exercise, one variable is expressed in terms of another and then replaced in the other equation(s) to reduce the system's complexity. The key is to find a single solution that works for all equations, which in the case of our exercise was the ordered pair \(2, 2\), satisfying both equations in the system.