Problem 60 - Algebra 1: Concepts and Skills

Question

Use linear combinations to solve the linear system. Then check your solution. $$ \begin{aligned} &4 x+5 y=7\\\ &6 x-2 y=-18 \end{aligned} $$

Step-by-Step Solution

Answer

The solution to the system of equations is x = -2 and y = 3.

Step 1: Identify the linear system

The linear system that needs to be solved is defined by two equations: \(4x + 5y = 7\) and \(6x - 2y = -18\).

Step 2: Select multipliers that will allow elimination one of the variables when the equations are combined

Multiply the first equation by 2 and the second equation by 5 to allow elimination of the variable y when we combine the two equations. The system becomes: \(8x + 10y = 14\) and \(30x - 10y = -90\).

Step 3: Combine the two equations to eliminate y

Add the two equations, \(8x + 10y = 14\) and \(30x - 10y = -90\). This addition eliminates y, resulting in the equation: \(38x = -76\).

Step 4: Find the value of x

Solve the equation from Step 3, \(38x = -76\), for x. To get the value of x, divide both sides of the equation by 38, therefore x = -2.

Step 5: Substitute x into the first equation

Substitute x = -2 into the first original equation, \(4x + 5y = 7\), resulting in the equation, \(-8 + 5y = 7\).

Step 6: Solve for y

To solve for y, add 8 to both sides of the equation, \(-8 + 5y = 7\), to get the expression, \(5y = 15\). Then, divide both sides of the equation by 5 to find the value of y, namely, y = 3.

Step 7: Checking the solution by substituting x and y into both equations

Sync the values x = -2 and y = 3 into the original equations \(4x + 5y = 7\) and \(6x - 2y = -18\), check that both left-hand sides equal to right-hand sides, to ensure the solution is correct.

Key Concepts

Linear Combinations

Linear combinations are a crucial tool for solving systems of equations. They involve adding or subtracting multiple equations to eliminate one variable. This simplifies the system to a more manageable form, allowing us to solve for the remaining variables. In the given exercise, the concept of linear combinations is used effectively. We multiplied each equation by a number to align the coefficients of one variable (in this case, variable \(y\)).
The process involves:

• Choosing a variable to eliminate.
• Adjusting one or both equations by multiplying them with a carefully chosen number.

By aligning the coefficients, we turn the system into a simpler form that allows for easy elimination of a variable, leading us to a single equation in one variable, just like reducing the system to the equation \(38x = -76\) in our example.

Elimination Method

The elimination method is an efficient way to find solutions to systems of equations. It's particularly useful when dealing with linear equations. This method simplifies the system by systematically eliminating variables. In our problem, we start by eliminating \(y\) through the clever use of linear combinations. This is done by:

• Multiplying the equations in the system to get opposite coefficients for \(y\).
• Adding the new equations to cancel out \(y\), leading to a single equation: \(38x = -76\).

After eliminating one variable, we solve for the remaining variable. This involves straightforward algebra, finding that \(x = -2\). The elimination of variables step by step reduces the complexity significantly, providing a clearer path to the solution.

Solution Verification

Verification is a vital step to ensure the correctness of your solution. Once you have potential values for \(x\) and \(y\), substitute them back into the original equations. This test ensures that both equations are satisfied, confirming the validity of the solution. In our case:

• Substitute \(x = -2\) and \(y = 3\) into \(4x + 5y = 7\).
• Verify that the resulting computation equals the equation's right side.
• Repeat for the second equation \(6x - 2y = -18\).

If both equations are true with the assigned variables, your solution is verified. Having a verified solution boosts confidence in the answer provided, proving that both steps and calculations are correct.

Systems of Equations

A system of equations contains several equations that share the same set of variables. Solving such a system means finding a common solution for all the equations involved. Each equation in the system provides a constraint that the solution must satisfy. In our scenario, the system is:

• \(4x + 5y = 7\)
• \(6x - 2y = -18\)

The solution of this system is the values of \(x\) and \(y\) that satisfy both equations simultaneously. In practice, systems of equations can occur in various forms, often requiring different techniques like substitution or elimination methods based on the complexity and structure of the equations. Understanding these systems helps solve real-world problems like coordinating schedules or predicting outcomes in engineering tasks.