Problem 13 - Thinking Mathematically

Question

In Exercises 13-24, solve each system by the substitution method. Be sure to check all proposed solutions. \(\left\\{\begin{array}{l}x+y=4 \\ y=3 x\end{array}\right.\)

Step-by-Step Solution

Answer

The solution to the given system of equations is \( x = 1 \) and \( y = 3 \)

Step 1: Substituting y from the second equation into the first equation

From the given system of equations, we know that \( y = 3x \). So we can substitute \( y \) in the first equation: \( x + (3x) = 4 \). Simplify this to obtain: \( 4x = 4 \).

Step 2: Solve for x

Now we can isolate \( x \) by dividing both sides of the equation by 4: \( x = 4 / 4 = 1 \). So the solution for \( x \) is 1.

Step 3: Solve for y

Substitute \( x = 1 \) into the equation \( y = 3x \) and calculate the corresponding \( y \): \( y = 3 * 1 = 3 \). So the solution for \( y \) is 3.

Step 4: Checking the solution

Substitute \( x = 1 \) and \( y = 3 \) into both original equations to ensure they hold true. For equation 1: \( 1 + 3 = 4 \). And for equation 2: \( y = 3 * 1 = 3 \). Both equations hold true, confirming that our solution is correct.

Key Concepts

Understanding Systems of Equations

A system of equations consists of two or more equations with the same set of variables. In the realm of algebra, understanding how to navigate through these systems is essential because it allows us to find the point at which two equations intersect, representing the solution that satisfies all equations involved.

In the case above, we worked with a simple system of two linear equations. The goal is to find the values of the variables, here noted as 'x' and 'y,' that make both equations true simultaneously. When graphed, each equation would correspond to a line, and the solution is the coordinate point where these lines intersect.

Algebraic Techniques for Equation Solving

Algebraic techniques for solving equations revolve around manipulating the equations to isolate the variable we're solving for. We use operations like addition, subtraction, multiplication, division, and substitution to rewrite the equations in a simpler form.

The substitution method, demonstrated in the original exercise, is particularly useful when one equation in the system can be easily solved for one variable in terms of the others. This expression for the variable is then substituted into the other equation(s), thus reducing the system's complexity and bringing us closer to the answer.

Solving Linear Equations

Solving linear equations is the foundation of algebra. 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 be written in various forms, but they'll always graph as straight lines.

To solve a linear equation, we aim to isolate the variable by performing operations that maintain the equation's balance. The process often involves combining like terms, which are terms that have the same variables raised to the same power, and then using the properties of equality to solve for the unknown variable. For example, in the step-by-step solution above, we simplified the equation to isolate 'x' and then calculated its value. Once 'x' was found, we used it to solve for 'y'. With both variables' values determined, we verified the solutions by plugging them back into the original equations.