Problem 39 - Introductory Algebra for College Students

Question

Multiply each equation in the system by an appropriate number so that the coefficients are integers. Then solve the system by the substitution method. $$\left\\{\begin{array}{l}0.7 x-0.1 y=0.6 \\ 0.8 x-0.3 y=-0.8\end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = 2\) and \(y = 8\).

Step 1: Multiply to Eliminate Decimals

Multiply the first equation by 10 and the second one by 10 as well to eliminate decimals. This gives the following system of equations:\[\begin{align*}7x - y &= 6, \\ 8x - 3y &= -8.\end{align*}

Step 2: Rearrange First Equation

Rearrange the first equation to isolate variable 'y'. This gives: \[y = 7x - 6.\]

Step 3: Substitute in Second Equation

Substitute this expression for 'y' in the second equation. This gives: \[8x - 3(7x - 6) = -8.\] Simplify this equation to get: \[-13x + 18 = -8.\]

Step 4: Solve for 'x'

Solve this equation for 'x' by first subtracting 18 from both sides, then dividing by -13. This gives: \[x = 2.\]

Step 5: Substitute 'x' in First Equation

Substitute this value for 'x' back into the first equation (after it was rearranged). This gives: \[y = 7*2 - 6 = 8.\]

Step 6: Solution

So, the solution to the system of equations is \(x = 2\) and \(y = 8\). These values satisfy both of the original equations.

Key Concepts

System of Linear Equations

A system of linear equations is a set of two or more linear equations involving the same set of variables. In simpler terms, it’s like having multiple lines drawn on a graph, and you need to find where they intersect, if they do at all. In our exercise, we are given a system with two equations and two variables, \(x\) and \(y\). Each equation in the system represents a line on a graph, and our goal is to find the values of \(x\) and \(y\) where these lines cross each other.

This intersection point usually provides the solution to the system of equations. Finding this solution means that both values satisfy all the equations in the system. Systems of linear equations are fundamental because they help us understand relationships between variables. They are used in various real-world situations such as calculating speeds, predicting trends, and solving problems related to resource allocation.

Elimination of Decimals

Eliminating decimals from mathematical expressions or equations helps in simplifying the calculations. When our equations include decimals, as seen in the exercise, it can make manipulation a little tricky. However, by multiplying each term by a power of 10, we can convert these decimals into integers, which are easier to handle.

In our example, each original equation was multiplied by 10, transforming the system into:

• \(7x - y = 6\)
• \(8x - 3y = -8\)

This process cleans up the problems involving decimals and lays down a clearer path for us to solve the equations. By transforming decimals into whole numbers, we make the equations more straightforward and approachable, paving the way for subsequent solving methods like substitution or elimination.

Solving for Variables

Solving for variables means finding the values of the unknowns that satisfy an equation. In the substitution method, we isolate one variable in one of the equations and then substitute that expression into the other equation. This approach allows us to reduce the system to a single equation with one variable, making it easier to solve.

In our exercise:

• We first rearranged the first equation to express \(y\) in terms of \(x\) (y = 7x - 6).
• Next, we substituted this expression into the second equation.
• Simplifying the resulting equation allowed us to find \(x = 2\).
• Finally, substituting \(x = 2\) back into the expression for \(y\), we found \(y = 8\).

This systematic approach ensures we efficiently and accurately find the values of \(x\) and \(y\), which together represent the solution to the system of equations. Understanding this method gives us a powerful tool for solving various mathematical problems.