Problem 109 - Algebra 1

Question

Use linear combinations to solve the system. (Review 7.3 ) $$\begin{aligned}&10 x-3 y=17\\\&-7 x+y=9\end{aligned}$$

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = -4\) and \(y = -19\)

Step 1: Multiply the equations

For the given system of equations \(10x - 3y = 17\) and \(-7x + y = 9\), it will be effective to multiply the second equation by 3, so that the coefficients for 'y' in both equations will cancel each other out when added. After multiplying the second equation by 3, the new equations to be solved are \(10x - 3y = 17\) and \(-21x + 3y = 27\)

Step 2: Add the equations

The aim now is to add the two equations from Step 1, such that the coefficients of 'y' will cancel out each other. This results in \(-11x = 44\)

Step 3: Solve for variable x

From Step 2, the variable x can be found by dividing the equation \(-11x = 44\) by -11. This gives \(x = -4\)

Step 4: Substitute x into second equation

Substitute this value of x into the second equation (\(-7x + y = 9\)), we get: \(-7(-4) + y = 9\), from which we find that \(y = -19\) after simplifying

Step 5: Verification

To verify these solutions, \(x = -4\) and \(y = -19\), should satisfy both the original equations \(10x - 3y = 17\) and \(-7x + y = 9\). By substituting these values into both equations, it is confirmed that they are solutions.

Key Concepts

System of Equations

When dealing with a system of equations, you're working with two or more equations that involve the same set of variables. Each equation presents a relationship between the variables, such as constants and coefficients. The goal is to find a common solution for all equations in the system, meaning the values of the variables satisfy each equation simultaneously.
To visually understand this, imagine each equation as a line on a graph. Solving the system is akin to finding the point or points where these lines intersect.

• If there is one solution, the lines intersect at a single point.
• If there are no solutions, the lines are parallel and never meet.
• If there are infinitely many solutions, the lines lie on top of each other completely overlapping.

Recognizing the nature of the solution set is crucial in understanding how different systems behave.

Solving Linear Equations

Solving linear equations is about finding the value of the variable that makes the equation true. Linear equations are relations of the first order. They tend to graph as straight lines on the coordinate plane.
In a system of linear equations, like the one in the exercise, we often solve by transforming and combining equations to isolate one variable. This typically follows a series of algebraic manipulations, including addition, subtraction, multiplication, or division of equations to align coefficients and eliminate variables.
For example, if given two equations, such as those from the exercise:

• Equation 1: \(10x - 3y = 17\)
• Equation 2: \(-7x + y = 9\)

We systematically adjust these equations to simplify and find the exact value of each variable.
The goal is to reduce the system to a single variable equation, which is straightforward to solve.

Substitution Method

The substitution method is a common strategy for solving systems of equations. It involves solving one of the equations for one variable and then substituting that expression into the other equation.
This method is particularly useful when the system results in simpler expressions for one variable. Here's how the substitution method works:

• First, solve one of the equations for one of its variables. This means isolating the variable on one side of the equation.
• Next, substitute this expression into the other equation. This will give you an equation with a single variable.
• Solve this new equation for the solitary variable.
• Finally, substitute that solution back into the first equation to solve for the remaining variable.

In the exercise, this step was inverted because solving through elimination was more efficient, which is another effective method closely related. However, understanding substitution remains valid and applicable for numerous systems and provides a foundational skill in algebra.