Problem 20 - 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.$$ \left\\{\begin{array}{c} 10 x+3 y=-12 \\ 5 x=-4 y-16 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \(x = 0\) and \(y = -4\).

Step 1: Rearrange the Second Equation

Let's first change the second equation to standard form. We have \(5x = -4y - 16\). To bring all terms to one side of the equation, add \(4y\) to both sides: \(5x + 4y = -16\).

Step 2: Align the System for Addition

Now our system looks like this: \(10x + 3y = -12\) and \(5x + 4y = -16\). Already aligned for the addition method. We'll adjust the second equation for addition by multiplying it by 2.

Step 3: Multiply the Second Equation

Multiply the entire second equation by 2 to make the coefficients of \(x\) equal. The equation \(5x + 4y = -16\) becomes \(10x + 8y = -32\).

Step 4: Add the Equations Together

Add the rewritten system: \(10x + 3y = -12\) and \(10x + 8y = -32\). The coefficients of \(x\) cancel: \((10x - 10x) + (3y + 8y) = -12 - 32\), resulting in \(11y = -44\).

Step 5: Solve for \(y\)

Divide both sides of the equation \(11y = -44\) by 11 to isolate \(y\): \(y = -4\).

Step 6: Substitute \(y\) in the First Original Equation

Substitute \(y = -4\) back into the first original equation: \(10x + 3(-4) = -12\), which simplifies to \(10x - 12 = -12\).

Step 7: Solve for \(x\)

Add 12 to both sides of the equation to isolate \(x\): \(10x = 0\). Divide both sides by 10: \(x = 0\).

Step 8: Verify the Solution

Substitute \(x = 0\) and \(y = -4\) back into the original equations to verify: \(10(0) + 3(-4) = -12\) and \(5(0) = -4(-4) - 16\), both are true, confirming the solution.

Key Concepts

Addition Method

The addition method, often referred to as the elimination method, is a technique used to solve systems of linear equations. This method is especially effective when you can easily eliminate one of the variables by adding or subtracting equations. To use this approach:

• Ensure that both equations are in a standard form, which means aligning the terms across both equations.
• Make the coefficients of one variable the same (or opposites) by multiplying one or both equations.
• Add or subtract the equations to eliminate one variable, simplifying the system to a single variable equation.

By eliminating a variable, solving systems of equations becomes more straightforward as it reduces the complexity to a single equation. In our example, after making the coefficients of \(x\) equal, adding the equations removed \(x\), allowing us to solve directly for \(y\).

Linear Equations

Linear equations are mathematical statements that depict a straight line when graphed on a Cartesian plane. They follow the general format \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. Linear equations are characterized by:

• Variables raised to the power of one, meaning no exponents greater than one.
• Graphing them produces a straight line, hence the name 'linear'.
• They're often the building blocks in a system of equations.

In our given problem, each equation in the system aligns with the structure of a linear equation. Solving these offers insight into how two quantities relate in a two-dimensional space.

Solving Equations

Solving equations involves finding the value(s) of the variable(s) that make the equation true. When dealing with a system of equations, you're seeking a set of values that satisfy all equations concurrently. Here’s the general approach:

• Use techniques like the substitution method or the addition method to eliminate variables.
• Reduce the system to one or more simpler equations.
• Solve these simpler equations to find the unknowns.

For our exercise, we successfully employed the addition method, transforming two equations into one, which was then easily solvable for \(y\). Substituting back into the original equations provided the remaining solution for \(x\). Verifying this solution ensured accuracy.

Mathematical Operations

Mathematical operations are the actions we perform on numbers or variables to achieve a desired result, and they are fundamental when solving systems of equations. The key operations include:

• Addition and Subtraction: Essential for both aligning equations and eliminating variables.
• Multiplication: Used to adjust coefficients to facilitate elimination.
• Division: Often the final step in isolating a variable for its solution.

In our solution, we primarily used addition to cancel out the \(x\) terms, and multiplication to adjust the coefficients. Division was used to solve the final simplified equation for \(y\). Understanding these operations allows for smoother problem-solving in linear systems.