Problem 25 - Algebra 1

Question

Use linear combinations to solve the system of linear equations. $$\begin{aligned} &x+3 y=12\\\ &-3 y+x=30 \end{aligned}$$

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = 21\) and \(y = -3\).

Step 1: Rearrange the Equations

The first step is to rearrange the equations in ascending order means to write the \(x\) terms first then \(y\) terms. In this case, the equations can be written as \[x + 3y = 12\] and \[x - 3y = 30\]

Step 2: Use the Linear Combination Method to Solve for One Variable

The linear combination method involves adding or subtracting the two equations to eliminate one of the variables. In this case, if we add the two equations, \[x + 3y + x - 3y = 12 + 30\], \(y\) terms cancel out, and we get \[2x = 42\]

Step 3: Solve the Equation for \(x\)

By further simplifying the equation from Step 2, we can solve for \(x\). We can do this by dividing both sides of the equation by 2. That gives us \[x = 21\]

Step 4: Substitute \(x\) into One of the Original Equations and Solve for \(y\)

By substituting \(x = 21\) into the first original equation \[x + 3y = 12\], we can solve for \(y\). Our equations becomes \[21 + 3y = 12\], which simplifies to \[3y = -9\], and further dividing by 3 gives us \[y = -3\]

Step 5: Check the Solution

It's always important to check the solution by substituting the variables \(x = 21\) and \(y = -3\) back into the original equations to ensure they satisfy both equations, that is \[x + 3y = 21 - 3*-3 = 21 + 9 = 30\] and \[x - 3y = 21 - 3*-3 = 21 + 9 = 30\], which is true, so the solution is correct.

Key Concepts

Linear Combinations

In a system of linear equations, linear combinations can be used to simplify the problem by eliminating one of the variables. This process involves adding or subtracting equations to cancel out a variable, making it easier to solve for the remaining variable. Imagine you have equations like

• \(x + 3y = 12\)
• \(x - 3y = 30\)

In this example, if you add the two equations, the \(y\) terms (+3y and -3y) cancel each other out. This leaves you with an equation solely in terms of \(x\). The power of using linear combinations lies in its ability to make systems of equations more approachable by reducing the number of unknowns involved in one of the equations. This is typically done by aligning terms of the same variable vertically first.

Substitution Method

Once linear combinations have simplified the problem, the substitution method can be used to further solve the system. After obtaining the value of one variable through linear combinations, substitute this value into one of the original equations to find the other variable.For instance, after determining \(x = 21\) by linear combinations, use substitution to solve for \(y\) as follows:

• Substitute \(x\) in the equation: \(x + 3y = 12\)
• Replace \(x\) with 21: \(21 + 3y = 12\)

This substitution transforms the remaining equation into something solvable, thus helping you find the value of \(y\). By engaging in substitution, you carry over the known value into the original equations, reducing complexity and solving for the next unknown.

Solving for Variables

The ultimate goal in solving any system of linear equations is to find the values of all variables involved. This process may involve a combination of methods, like linear combinations and substitution. After reducing the system into simpler equations, focus on isolating variables on one side of the equation. For example, with \(3y = -9\), solve for \(y\) by dividing both sides by 3, giving you \(y = -3\).

• Ensure each equation results in an isolated variable
• Use arithmetic operations: addition, subtraction, multiplication, or division

Checking your solution by substituting back into the original equations can confirm your findings are correct. Ensuring correctness is crucial in systems where multiple variables interdependently satisfy each equation. By practicing solving for variables step-by-step, confidence and proficiency in these techniques are both bolstered.