Problem 6 - College Algebra

Question

Solve each system by the substitution method. \(\left\\{\begin{array}{l}x+y=6 \\ y=2 x\end{array}\right.\)

Step-by-Step Solution

Answer

The solution to the system of equations is \(x=2\) and \(y=4\).

Step 1: Substitution

Substitute 'y' from the second equation into the first equation to get a new equation. This gives: \(x+2x=6\).

Step 2: Combine like terms

Next, simplify the equation by combining the terms with the variable 'x'. This results in: \(3x=6\).

Step 3: Solve for x

Now, the equation \(3x=6\) can be solved for 'x' by dividing both sides by 3. This gives: \(x=2\).

Step 4: Solve for y

By substituting \(x=2\) into the second equation \(y=2x\), we can solve for 'y'. This gives: \(y=2*2=4\).

Key Concepts

Systems of Equations

A system of equations consists of two or more equations that share common variables. The main goal is to find the values of these variables that satisfy all given equations in the system simultaneously. In most problems, equations involve a mix of letters (representing variables) and numbers, and each equation can represent a line, curve, or other geometric figure. They are often used in various fields such as mathematics, physics, and economics to model real-world situations.

• One way to solve systems is by using the substitution method, where you solve one equation for one variable and substitute that expression into another equation.
• Another common method is the elimination method, where you add or subtract equations to eliminate a variable.

Substitution is particularly handy when one of the equations is already solved for a single variable or can be manipulated easily to express one variable in terms of another. This leads us to our next concept: algebraic solutions.

Algebraic Solutions

Algebraic solutions involve finding the exact values of variables that satisfy an equation or a system of equations. Instead of using graphical methods, which can be approximate and tedious for complex systems, algebraic methods provide a precise solution.

• The substitution method is a core algebraic technique for solving equations and works well for systems of two variables.
• It involves making one of the variables the subject in one of the equations and then replacing it in the other equation(s) in the system. This results in a single equation with one variable.

By solving this resulting single-variable equation, you find the value of one variable, which can then be substituted back into original equations to find the values of remaining variables. This transition from two variables to one variable equation simplifies solving the system greatly.

Solving for Variables

Solving for variables means isolating a variable on one side of the equation to determine its value. The idea is to manipulate the equation, using various algebraic techniques, until the variable of interest stands alone on one side of the equal sign.

• In a system of equations, once you have a single equation in one variable, you solve it like any other basic algebraic equation.
• For example, if you have the equation \(3x=6\), you divide both sides by 3 to isolate \(x\), which gives \(x=2\).

Once one variable is found, substituting it back into one of the original equations allows you to solve for the next variable. With efficient techniques like substitution, solving for variables in a system of equations becomes a systematic and logical process. This ensures that you're tackling each variable one at a time, making the problem simpler as you proceed.