Problem 16 - Intermediate Algebra

Question

For Problems \(1-18\), use the elimination-by-addition method to solve each system. (Objective 1 ) $$ \left(\begin{array}{r} 10 x-8 y=-11 \\ 8 x+4 y=-1 \end{array}\right) $$

Step-by-Step Solution

Answer

The solution is \(x = -\frac{1}{2}\) and \(y = \frac{3}{4}\).

Step 1: Align the Equations

Begin by writing the system of equations in column form for clarity:\[ \begin{align*} 10x - 8y &= -11 \ 8x + 4y &= -1 \end{align*} \]

Step 2: Adjust the Equations

We will aim to eliminate one of the variables by ensuring the coefficients of either \(x\) or \(y\) are equal in absolute value for both equations. In this case, multiply the second equation by 2:\[ \begin{align*} 10x - 8y &= -11 \ 2(8x + 4y) &= 2(-1) \end{align*} \]which simplifies to \[ \begin{align*} 10x - 8y &= -11 \ 16x + 8y &= -2 \end{align*} \]

Step 3: Add the Equations

Add the two equations together to eliminate \(y\):\[ \begin{align*} (10x - 8y) + (16x + 8y) &= -11 + (-2) \ 26x &= -13 \end{align*} \]

Step 4: Solve for x

Divide both sides of the equation by 26 to solve for \(x\):\[ x = \frac{-13}{26} \]Thus, \[ x = -\frac{1}{2} \]

Step 5: Substitute to Find y

Substitute \(x = -\frac{1}{2}\) back into one of the original equations, such as the first one:\[ 10(-\frac{1}{2}) - 8y = -11 \]This simplifies to:\[ -5 - 8y = -11 \]Add 5 to both sides:\[ -8y = -6 \]

Step 6: Solve for y

Divide both sides by -8 to find \(y\):\[ y = \frac{-6}{-8} \]Thus, \[ y = \frac{3}{4} \]

Step 7: Verify the Solution

Substitute \(x = -\frac{1}{2}\) and \(y = \frac{3}{4}\) into the second original equation to verify:\[ 8(-\frac{1}{2}) + 4(\frac{3}{4}) = -1 \]This simplifies:\[ -4 + 3 = -1 \]Both sides are equal, confirming the solution is correct.

Key Concepts

Systems of Equations

A system of equations is a collection of two or more equations with the same set of variables. In simple terms, it's like a couple of math problems that you need to solve together, not separately. Each equation in the system represents a line if you were to graph it on a plane.

Imagine you have two equations:

• 10x - 8y = -11
• 8x + 4y = -1

These lines might intersect at a specific point, which is the solution to the system. The goal when dealing with systems of equations is to find the values for the variables that make both equations true at the same time.

In this exercise, we use a powerful technique called the elimination method to simplify our work. This is one common method, among others like graphing and substitution, to solve systems of equations. The elimination method focuses on removing one of the variables to make solving as straightforward as possible.

Linear Equations

Linear equations are at the heart of systems of equations. These are equations that, when graphed, yield straight lines. The term 'linear' indicates that each equation correlates to a straight line in the coordinate plane.

A linear equation typically looks like this: \[ ax + by = c \]where the \(a\), \(b\), and \(c\) are constants, and \(x\) and \(y\) are variables. In our specific example:

• 10x - 8y = -11
• 8x + 4y = -1

These equations will graph into lines that may intersect, and the solution is where they meet. The beauty of linear equations in a system is their predictability: two equations, two variables, and they can have:

• One solution, where they intersect once.
• No solution, if they are parallel.
• Infinite solutions, if they are the same line.

In this method, we aim to turn these linear equations into a simpler form by making the lines easier to manage through elimination.

Algebraic Solutions

The objective of algebraic solutions is to find numerical or symbolic answers to equations using algebraic manipulations. The elimination method is particularly well-suited for algebraic solutions because it leverages arithmetic manipulations to simplify solving systems of equations.

In our problem, we started by aligning the equations. Then, we adjusted them to make one variable have the same coefficient in both equations but with opposite signs. For instance, we multiplied the second equation by 2 to get:\[ 16x + 8y = -2 \]Next, by adding the adjusted equations together:\[ (10x - 8y) + (16x + 8y) = -11 + (-2) \]we eliminated \(y\) from the system, which simplified the process to solving a single equation in terms of \(x\). Once \(x\) was found as \(-\frac{1}{2}\), substituting it back helps find \(y\) as \(\frac{3}{4}\).

Verification of these values in the original equations confirmed the correctness of the solution. This process showcases the power of algebra to directly solve complex questions with straightforward calculations and logical steps.