Problem 31 - Algebra A Combined Function

Question

Solve each system of equations by the addition method. If a system contains fractions or decimals, you may want to first clear each equation of fractions or decimals.$$ \left\\{\begin{array}{l} x-\frac{y}{3}=-1 \\ -\frac{x}{2}+\frac{y}{8}=\frac{1}{4} \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \(x = 1\) and \(y = 6\).

Step 1: Eliminate Fractions

First, clear the equations of fractions by multiplying each term by the least common multiple of the denominators. For the first equation, the denominators are 1 and 3. Multiply the entire equation by 3: \[ 3(x - \frac{y}{3}) = 3(-1) \] This simplifies to:\[ 3x - y = -3 \]For the second equation, the denominators are 2 and 8. Multiply the entire equation by 8:\[ 8(-\frac{x}{2} + \frac{y}{8}) = 8(\frac{1}{4}) \]This simplifies to:\[ -4x + y = 2 \].

Step 2: Add the Equations

Now, add the simplified equations from Step 1 together to eliminate one of the variables. Add:\[(3x - y) + (-4x + y) = -3 + 2\]This results in:\[-x = -1 \].

Step 3: Solve for the First Variable

Solve \(-x = -1\). Multiply both sides by -1 to find \(x\):\[ x = 1 \].

Step 4: Substitute to Find Second Variable

Substitute \(x = 1\) back into one of the original equations to solve for \(y\). Use the equation \(3x - y = -3\):\[ 3(1) - y = -3 \]This simplifies to:\[ 3 - y = -3 \]Add \(y\) to both sides and then subtract 3 from both sides:\[ y = 6 \].

Key Concepts

Addition Method

The addition method is a common technique for solving systems of equations. It's also called the "elimination method" because it allows us to eliminate one of the variables by adding the equations together. This method helps in simplifying the system and making it easier to solve.

• Start by aligning the equations vertically.
• Add the equations directly if the coefficients of one of the variables are opposite in sign and equal in magnitude. If not, you may need to multiply one or both equations by some numbers to make this happen.

In our exercise, once the equations were cleared of fractions, we could add them directly. This eliminated the variable "y," leaving only "x" to solve.

Eliminate Fractions

Eliminating fractions from equations is a crucial first step in solving equations using the addition method. Fractions can complicate calculations but can be cleared by multiplying each term in the equation by the least common multiple (LCM) of the denominators. To do this:

• Identify the denominators in each equation.
• Find the least common multiple (LCM) for these denominators.
• Multiply every term in the equation by this LCM.

After this operation, your equations will have integer coefficients, which simplifies further operations. In the given exercise, we multiplied the terms of the equations by 3 and 8, respectively, to clear the fractions.

Linear Equations

Linear equations represent lines on the coordinate plane and have the general form: \( ax + by = c \) where \(a\), \(b\), and \(c\) are constants.

• These equations graph as straight lines.
• The solution to a system of linear equations is the point where the lines intersect.

In a system of linear equations with fractions like in our exercise, solving becomes easier after eliminating fractions. Once in a simplified form, you can proceed with the addition method to find the point of intersection.

Substitution

The substitution method is another technique used in solving systems of equations. This method involves solving one of the equations for one variable and then substituting that expression into the other equation.

• Start by solving one equation for one of the variables.
• Use the solution to replace the variable in the second equation.
• Solve for the remaining variable.

In our exercise, after finding \( x = 1 \) using the addition method, we substitute this back into one of the simplified original equations to find \( y \). This approach complements the addition method by simplifying the system after some initial work.