Problem 38 - Algebra 2

Question

Solve each system of equations. $$ \left\\{\begin{array}{l}{2 x+2 y=10} \\ {3 x-y=4}\end{array}\right. $$

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = 2.25\) and \(y = 2.75\).

Step 1: Solve for y in the second equation

We start by isolating y in the second equation. We can do this by adding y to both sides and then subtracting 4 from both sides: \(3x - y = 4\) becomes \(y = 3x - 4\).

Step 2: Substitute y into the first equation

Next, substitute the expression for y from Step 1 into the first equation where y is located: \(2x + 2(3x - 4) = 10\).

Step 3: Solve for x

Simplify the equation from Step 2 and solve for x: \(2x + 6x - 8 = 10\) resulting in \(8x = 18\), so \(x = \frac{18}{8}\) which simplifies to \(x = 2.25\).

Step 4: Solve for y

Now that we have the value for x, we substitute it back into the equation we derived for y in Step 1: \(y = 3(2.25) - 4\), simplifying to \(y = 6.75 - 4\), which means \(y = 2.75\).

Key Concepts

Substitution Method

The substitution method is a straightforward technique used to solve systems of linear equations, where one variable is expressed in terms of the other. To begin, you take one of the equations in the system and solve it for one variable. Once you have isolated this variable, you can 'substitute' this expression into the other equation. This substitution will give you an equation with only one variable, which is much easier to solve.

In the example given, we started by isolating the variable y in the second equation. After finding that y could be expressed as \( y = 3x - 4 \), we substituted this expression for y into the first equation. The process eliminates the y variable, giving us an equation in only x, which we can solve. After finding x, we substitute it back to find y. This clarity and symmetry make the substitution method particularly popular for solving two-variable systems.

• Isolate one variable
• Substitute the isolated variable into the other equation
• Solve for the remaining variable
• Back-substitute to find the other variable

Linear Equations

Linear equations form the basis for the systems we are solving. A linear equation represents a straight line when graphed on a coordinate plane and is typically presented in the form \( ax + by = c \), where a, b, and c are constants, and x and y are variables. Solving linear equations involves finding the values of x and y that make the equation true.

The system of equations from the exercise consists of two linear equations. Systems of linear equations can intersect at a single point, which represents the solution to the system. The goal is to find the point (x, y) where both lines intersect. In the provided example, the intersection point is \( (x, y) = (2.25, 2.75) \), which satisfies both equations. This singularity of solution is a hallmark of systems of linear equations that are independent and consistent.

• Represents a straight line on the graph
• Often written as \( ax + by = c \)
• Solutions are points where lines intersect

Algebraic Methods

Algebraic methods, including the substitution method, encompass various techniques employed to find solutions to equations. Other strategies to solve systems of equations include the elimination method, where you add or subtract equations to eliminate one of the variables, and the graphical method, where you find the intersection of lines on a graph.

Deciding which algebraic method to use can depend on the complexity of the system and personal preference. The overall goal of these methods is to reduce the system of equations to a simpler form that can easily be solved for the unknown variables. Each method has its own advantages, and for many students, the substitution method often provides a clear path to the solution. Nonetheless, it is essential to be comfortable with multiple methods to address a wide range of problems effectively.

• Diverse techniques to solve equations
• Includes substitution, elimination, and graphical methods
• Each method has unique advantages