Problem 55 - Algebra 2

Question

PREREQUISITE SKILL. Solve each system of equations. $$ \begin{array}{l}{x+y=10} \\ {2 x+y=11}\end{array} $$

Step-by-Step Solution

Answer

The solution is \( x = 1 \) and \( y = 9 \).

Step 1: Assign Equation Labels

Label the equations for clarity. Let Equation 1 be \( x + y = 10 \) and Equation 2 be \( 2x + y = 11 \).

Step 2: Subtract to Eliminate Variable

Subtract Equation 1 from Equation 2 to eliminate \( y \). This gives \((2x + y) - (x + y) = 11 - 10\), simplifying to \( x = 1 \).

Step 3: Substitute back to Find Second Variable

Substitute \( x = 1 \) into Equation 1. This gives \( 1 + y = 10 \). Simplifying, we get \( y = 9 \).

Step 4: Check the Solution

Substitute \( x = 1 \) and \( y = 9 \) back into both original equations to ensure they hold true. For Equation 1: \( 1 + 9 = 10 \). For Equation 2: \( 2(1) + 9 = 11 \). Both are verified as true.

Key Concepts

Understanding the Elimination Method

The elimination method is a strategic approach used to solve systems of linear equations. It involves removing one variable, which simplifies the solving process. This method is particularly useful when the coefficients of one variable in two equations are the same or can easily be manipulated to be the same.
You eliminate a variable by adding or subtracting the equations from each other to cancel that variable out. Here's a brief rundown on how it works:

• Align the equations one beneath the other, ensuring that like terms are in the same column.
• Decide which variable you want to eliminate first, and modify the equations if needed to ensure the coefficients are matching.
• Add or subtract the equations to cancel out one variable.

By removing one of the unknowns, you are left with a single-variable linear equation, which can be solved easily. Once the value of one variable is determined, substitute it back into one of the original equations to find the value of the other variable.

Exploring the Substitution Method

The substitution method is another effective technique for solving systems of linear equations. This approach is about solving one equation for one variable and then substituting the derived expression into the other equation.
This method is quite effective when one of the equations is already arranged to quickly express one variable in terms of the other. Here's how the substitution method simplifies the process of solving linear equations:

• Solve one equation for one of the variables, which makes it easier to handle.
• Take the expression obtained from this step and substitute it into the other equation.
• This substitution will give you an equation with one variable which you can solve directly.

After finding the value of one variable, back-substitute into one of the original equations to solve for the other variable. This method is especially useful for equations that are already set up to make expressing one variable in terms of the other straightforward.

The Basics of Linear Equations

Linear equations are equations of the first degree, meaning the highest power of the variable is one. They are called "linear" because their graphical representation is a straight line. The general form of a linear equation in two variables, such as our problem, is \(ax + by = c\).
Linear equations can have one, none, or infinitely many solutions. Solving linear equations is foundational, as they are easier to handle and form the building blocks for more complex mathematical concepts. Here's a quick look at some characteristics:

• Each equation represents a line in a plane.
• Solving a system of linear equations involves finding where these lines intersect.
• Intersections provide the set of solutions that satisfy all equations simultaneously.

In practice, systems of linear equations often model real-world situations where two or more conditions must be met concurrently. Understanding their solutions helps in applications ranging from economics to engineering.