Question

Solve each system of equations. If the system has no solution, state that it is inconsistent. $$ \left\\{\begin{array}{l} \frac{1}{3} x-\frac{3}{2} y=-5 \\ \frac{3}{4} x+\frac{1}{3} y=11 \end{array}\right. $$

Short answer

The solution is \( x = 12 \) and \( y = 6 \).

Step-by-step solution

- Eliminate fractions

Multiply both sides of each equation by the least common multiple (LCM) of the denominators to eliminate the fractions: First equation: Multiply by 6 (LCM of 3 and 2)\[ 6 \left( \frac{1}{3} x - \frac{3}{2} y \right) = 6(-5) \]This simplifies to: \[ 2x - 9y = -30 \]Second equation: Multiply by 12 (LCM of 4 and 3)\[ 12 \left( \frac{3}{4} x + \frac{1}{3} y \right) = 12(11) \]This simplifies to: \[ 9x + 4y = 132 \]

- Align the system

Write the simplified system of equations:\[ \begin{cases} 2x - 9y = -30 \ 9x + 4y = 132 \end{cases} \]

- Multiply to align coefficients

To eliminate one variable, we align the coefficients of either variable by multiplying each equation as needed. For this case, multiply the first equation by 9 and the second equation by 2:First equation: \[ 9(2x - 9y) = 9(-30) \]This becomes: \[ 18x - 81y = -270 \]Second equation: \[ 2(9x + 4y) = 2(132) \]This becomes: \[ 18x + 8y = 264 \]

- Subtract equations to eliminate a variable

Subtract the first modified equation from the second to eliminate \(x\):\[ (18x + 8y) - (18x - 81y) = 264 - (-270) \]This simplifies to: \[ 89y = 534 \]Solve for \(y\):\[ y = \frac{534}{89} = 6 \]

- Substitute back to find \(x\)

Substitute \(y = 6\) back into one of the original equations. Use \(2x - 9y = -30\):\[ 2x - 9(6) = -30 \]This simplifies to: \[ 2x - 54 = -30 \]Add 54 to both sides:\[ 2x = 24 \]Solve for \(x\):\[ x = 12 \]

Key concepts

elimination method

The elimination method is a powerful technique for solving systems of linear equations. It involves combining the equations to eliminate one of the variables, making it easier to find the values of the remaining variables. This method is particularly useful when the coefficients of the variables can be easily manipulated to cancel each other out.

• Step 1: Multiply each equation by a suitable number to make the coefficients of one of the variables equal (or opposites).
• Step 2: Add or subtract the equations to eliminate one variable.
• Step 3: Solve the resulting single-variable equation.
• Step 4: Substitute the found value back into one of the original equations to find the other variable.

By following these steps, the elimination method simplifies the process of solving complex systems. It’s particularly effective when one can quickly align the coefficients of one variable.

least common multiple

The least common multiple (LCM) is crucial in the elimination method, especially when dealing with fractions. The LCM is the smallest number that is a multiple of all denominators involved. Multiplying each term in the equation by the LCM helps to clear fractions, simplifying the system.Consider the exercise:

• For the first equation, \( \frac{1}{3} x - \frac{3}{2} y = -5 \), the LCM of 3 and 2 is 6.
• For the second equation, \( \frac{3}{4} x + \frac{1}{3} y = 11 \), the LCM of 4 and 3 is 12.

After multiplying, the equations become fraction-free, making further steps easier and reducing potential errors.

substitution method

The substitution method is another technique used to solve systems of equations. Unlike elimination, it involves solving one equation for one variable and then substituting that expression into the other equation.Here’s how it works:

• Step 1: Solve one of the equations for one variable.
• Step 2: Substitute this expression into the other equation to solve for the second variable.
• Step 3: Substitute the second variable's value back into the first equation to find the first variable's value.

This method is particularly useful when one of the equations is already solved for one variable or can be easily manipulated to do so. Its simplicity makes it a preferred choice for systems that are not too complex.

algebra

Understanding algebra is fundamental to solving systems of equations using both the elimination and substitution methods. Algebra involves working with variables and constants to express relationships and solve equations.Key algebraic techniques include:

• Isolating variables by adding, subtracting, multiplying, or dividing both sides of an equation.
• Using properties like the distributive, associative, and commutative properties to simplify expressions.
• Substituting known values to find unknown variables.

In this exercise, algebraic manipulation is essential at each step—from clearing fractions using the LCM, aligning coefficients, and solving the simplified equations. Mastery of algebraic fundamentals enhances problem-solving skills and makes tackling complex systems more manageable.