Problem 39 - Algebra 2

Question

Solve each system by elimination. \(\left\\{\begin{array}{l}{y=4-x} \\ {3 x+y=6}\end{array}\right.\)

Step-by-Step Solution

Answer

The solution to the system of equations is x = 1, y = 3.

Step 1: Express one variable in terms of the other

The first equation gives us directly that y is expressed in terms of x: y = 4 - x.

Step 2: Substitute the expression for y into the second equation

Substitute y = 4 - x into the second equation: 3x + (4 - x) = 6.

Step 3: Simplify and solve for x

Combine like terms in the second equation to get 2x + 4 = 6, then subtract 4 from both sides to obtain 2x = 2. Finally, divide both sides by 2 to find x equals 1: x = 1.

Step 4: Substitute the value of x into the first equation to solve for y

With x = 1, substitute into the first equation to get y = 4 - 1, so y equals 3: y = 3.

Key Concepts

Algebraic Elimination Method

When it comes to solving systems of equations, the algebraic elimination method is one of the most straightforward techniques to use. It involves eliminating one of the variables so you can solve for the other. In practice, this means manipulating the equations by adding, subtracting, or multiplying them in order to cancel out one of the variables.

Let's walk through the use of the elimination method with an example:

• Consider the system of equations:
  \(y = 4 - x\)
  \(3x + y = 6\).
• To eliminate a variable, we aim to have the coefficients of either x or y be opposites in each equation. In this case, we can see that the coefficient of x in the first equation is -1 and in the second equation it is 3. This suggests that we could multiply the first equation by 3 to have coefficients that can cancel each other out on addition.
• After doing so, we would get \(3y = 12 - 3x\). Now when we add this to the second equation, the x terms will cancel: \((3x + y) + (12 - 3x) = 6 + 12\), simplifying to \(y = 18\). However, in this problem, the elimination process was even more straightforward because the equations were set up for immediate cancellation without the need for multiplication.

While the method is useful, it's not always immediately apparent how to manipulate the equations to cancel one of the variables. Sometimes, it requires multiplying both equations by different numbers to get the coefficients to align properly. Once a variable is eliminated and one of the equations is solved for the remaining variable, the solution can be substituted back into one of the original equations to find the value of the other variable.

Substitution Method

The substitution method is another powerful approach used to solve systems of linear equations. It involves solving one of the equations for one variable and then substituting that expression into the other equation. This process reduces the system to a single equation in one variable, which is generally easier to solve.

Here's how the substitution method works using the same example:

• First, solve one of the equations for one variable - the first equation already gives us \(y = 4 - x\).
• Next, substitute this expression into the other equation. In this case, we replace 'y' in the second equation with \(4 - x\), leading to \(3x + (4 - x) = 6\).
• Then, proceed to simplify and solve the new equation for 'x' just as done in the original step-by-step solution.
• Once we find 'x', we substitute it back into the expression we found for 'y' to get the value of 'y'.

This method is particularly useful when one of the equations is already solved for a variable, or can be easily manipulated to be so. An advantage of the substitution method is that it can be more intuitive, as you are directly inserting values from one equation into another. Always remember to check your solution by substituting the values back into the original equations to ensure they satisfy both.

Linear Equations

Linear equations are the foundation of algebra and represent the relationships where the variables are never multiplied together and don't have exponents other than 1. They can be easily recognized by their standard form \(ax + by = c\), where 'a', 'b', and 'c' are constants. In a system of linear equations, we're working with two or more of these equations at the same time, with the goal of finding the values for x and y that make all the equations true simultaneously.

For instance, consider the system from our exercise:

• The first equation, \(y = 4 - x\), is already in a simplified form and shows a direct relationship between 'y' and 'x'.
• The second equation, \(3x + y = 6\), is in standard form and demonstrates how both variables influence the outcome jointly.

Working with linear equations often involves graphing them on a coordinate plane. Each equation corresponds to a straight line, and the point where the two lines intersect represents the solution to the system. This geometric interpretation complements the algebraic methods of elimination and substitution, providing a visual way to understand the concept of finding a common solution.