Problem 30 - Algebra 1: Concepts and Skills

Question

Use linear combinations to solve the linear system. Then check your solution. \(5 s+8 t=70\) \(60=5 s-8 t\)

Step-by-Step Solution

Answer

The solution to the system of equations is \(s = 13\) and \(t = 5/8\).

Step 1: Add the Equations

By adding the two equations together, we can see that the term with variable \(t\) will be eliminated. This is because the term in the first equation is \(8t\) and in the second equation is \(-8t\). Thus, \(5s+8t+5s-8t=70+60\) simplifies to \(10s = 130\).

Step 2: Solve for First Variable

Solving the equation from step 1 for \(s\) gives us the first variable. We divide both sides of the equation \(10s = 130\) by 10 to isolate \(s\), resulting in \(s = 13\).

Step 3: Substitute and Solve for Second Variable

Now we will substitute \(s = 13\) into the first original equation to solve for \(t\). This results in \(5*13 + 8t = 70\), which simplifies to \(65 + 8t = 70\). Solving this equation for \(t\) by subtracting 65 from both sides and dividing by 8 gives us \(t = 5/8\).

Step 4: Check the Solution

We will confirm our solution by substituting \(s = 13\) and \(t = 5/8\) into both original equations. Substituting into the first equation gives us \(5*13 + 8*5/8 = 70\), which simplifies to \(70 = 70\), confirming the first equation. Substituting into the second equation gives us \(60 = 5*13 - 8*5/8\), which simplifies to \(60 = 60\) confirming the second equation. Thus our solution is correct.

Key Concepts

Linear Combinations

The method of linear combinations, also known as the addition method, is a strategy used to solve systems of linear equations. This technique involves combining the equations of the system in a way that allows us to eliminate one of the variables, making it easier to solve for the remaining variable. For example, consider the system from our exercise:

• 5s + 8t = 70
• 60 = 5s - 8t

When we add the two equations together, we eliminate 't' because we have a positive and negative coefficient of the same absolute value, which nicely cancels out the 't' terms. It's like strategically stacking blocks to make some of them vanish, thus simplifying our problem to just one variable.

System of Equations

A system of equations is a set of two or more equations with the same variables. The goal when solving a system of equations is to find the values for each variable that satisfy all equations in the system simultaneously. In the context of our exercise, we have a system of two equations with two variables, 's' and 't'. Systems can be solved using various methods, including graphing, substitution, elimination (linear combinations), or matrices. Audiences from different mathematical backgrounds might prefer different methods; however, the linear combinations method provides a straightforward approach to problem-solving when the coefficients of the variables line up favorably, as they do in our exercise.

Substitution Method

The substitution method is another powerful tool for solving systems of equations. This method involves solving one of the equations for one of its variables, and then 'substituting' that expression into the other equation. This transforms the system into a single equation in one variable, which is much easier to solve. For instance, after using linear combinations to find that s = 13, we then 'substituted' s in one of the original equations to find t. By plugging 13 in for s, we were left with an equation with only t, which we then could solve easily. It's a bit like solving a puzzle: find the right piece and everything else falls into place.

Check the Solution

After finding the solutions to the system of equations, it's crucial to check that these solutions actually work. Checking the solution involves substituting the values obtained back into the original equations to ensure that they satisfy each equation. In our exercise, substituting s = 13 and t = 5/8 back into the original equations showed that both of them hold true. This step is akin to taste-testing a dish after following the recipe – it confirms that you've cooked up the right results. It also serves as a verification method to prevent any undetected mistakes during calculations, ensuring the solution's accuracy and the student's understanding of the process.