Problem 29 - Precalculus with Limits

Question

Solving a System by Elimination In Exercises \(13-30\) , solve the system by the method of elimination and check any solutions algebraically. $$ \left\\{\begin{aligned} \frac{x+3}{4}+\frac{y-1}{3} &=1 \\ 2 x-y &=12 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The solution to the system is \(x = 5\) and \(y = -2\).

Step 1: Manipulate the first equation to eliminate the fractions

Multiply the first equation by 12 to get rid of the fractions.\nThis gives: \[3(x+3) + 4(y-1) = 12\]. Simplify to obtain: \[3x+9 + 4y-4 = 12\] or, \[3x + 4y + 5 = 12\]. Hence, the equation is: \[3x + 4y = 7\].

Step 2: Multiply the equations to align coefficients for elimination

Multiply the first equation by 2 and the second equation by 3 to get: \[6x + 8y = 14\] and \[6x - 3y = 36\].

Step 3: Solve the new system

Subtract the second equation from the first to get: \[11y = -22\]. Divide both sides by 11 to get: \[y = -2\].

Step 4: Substitute the solution to the second equation

Substitute \(y = -2\) into the second equation: \[2x - (-2) = 12\]. So, \[2x +2 = 12\], then \[2x = 10\] and finally, \[x = 5\].

Step 5: Check the solution algebraically

Substitute \(x = 5\) and \(y = -2\) into the original system to verify if the solutions satisfy both equations.

Key Concepts

Elimination Method

The elimination method is a powerful tool to solve systems of equations. Its main idea is to eliminate one variable by adding or subtracting equations, making it easier to solve for the remaining variable.
In the given exercise, we have two equations. To eliminate fractions in the first equation, we multiply it by 12, which is a common multiple of the denominators. This helps to convert the fractions into whole numbers, simplifying our calculations.
After manipulation, both equations are set up such that their coefficients of one variable can become equal. In this case, by multiplying the entire equations by appropriate values, the coefficients of \(x\) become 6. We can now subtract one equation from the other, which eliminates \(x\), leading us closer to our solution.

Algebraic Solution

Algebraic solutions involve logical manipulation of equations to find unknown variables.
Once we've eliminated fractions and aligned the coefficients, we subtract the equations to remove one variable. This process is methodical and relies on understanding operations like addition, subtraction, and multiplication of entire equations.
Although it might seem complex, the algebraic solution provides clarity and is rooted in simple arithmetic. After finding \(y = -2\), this value is substituted back into one of the original equations to solve for \(x\). This substitution ensures the discovered values satisfy both original equations, confirming the solution's accuracy.

Fractions in Equations

Fractions can make equations appear more complicated due to their denominators.
The key to handling fractions in systems of equations is to clear them out by multiplying through by a common multiple of the denominators. This transforms each term into whole numbers or decimals, making them easier to work with effectively.
In our problem, by multiplying the entire first equation by 12, the fractions are eliminated. This step simplifies the system, allowing us to focus on core algebraic manipulation without the distraction of fractional values. Understanding and executing this step smoothly is essential, especially in more complex problems.

Equation Manipulation

Manipulating equations involves using arithmetic operations to transform the equations into a form that is easier to solve. This might involve scaling equations or aligning coefficients for elimination.
In the solution provided, the manipulation includes both eliminating fractions and strategically multiplying entire equations. This is to get matching coefficients on the variables we wish to eliminate.

• First, fractions are removed.
• Second, equations are multiplied to align coefficients.
• Later, one equation is subtracted from the other to eliminate a variable.

Mastering equation manipulation provides a fundamental skill that simplifies seemingly complex systems, making them much more accessible. By practicing these methods, solving any system of equations becomes a straightforward task.