Problem 11 - Algebra Form and Function

Question

Solve the systems of equations. $$ \left\\{\begin{array}{l} 20 n+50 m=15 \\ 70 m+30 n=22 \end{array}\right. $$

Step-by-Step Solution

Answer

Answer: The values that satisfy both equations are m = 1/10 and n = 1/2.

Step 1: Isolate one variable in one equation for substitution

We will isolate the variable n in the first equation. $$20n + 50m = 15$$ Subtracting 50m from both sides, $$20n = 15 - 50m$$ Dividing both sides by 20, $$n = \frac{15 - 50m}{20}$$ Now we have n isolated and we are ready to substitute it into the second equation.

Step 2: Substitute the isolated variable into the second equation

Now, we substitute the expression for n from Step 1 into the second equation: $$70m + 30(\frac{15 - 50m}{20}) = 22$$

Step 3: Solve for the remaining variable

We will now solve for m in the equation we derived in Step 2: $$70m + \frac{30(15 - 50m)}{20} = 22$$ Multiply both sides by 20 to eliminate the fraction: $$20(70m) + 30(15 - 50m) = 22 * 20$$ Distribute: $$1400m + 450 - 1500m = 440$$ Combine like terms: $$-100m = -10$$ Divide by -100 to find m: $$m =\frac{-10}{-100}$$ So, $$m = \frac{1}{10}$$

Step 4: Substitute the value of m back into the equation for n

Now that we found the value of m, we can substitute it back into the equation for n from Step 1: $$n = \frac{15 - 50(\frac{1}{10})}{20}$$ Simplify: $$n = \frac{15 - 5}{20}$$ Divide: $$n = \frac{10}{20}$$ So, $$n = \frac{1}{2}$$

Step 5: Write the solution as an ordered pair

The solution to the system of equations is the ordered pair (m, n): $$(m, n) = \left(\frac{1}{10}, \frac{1}{2}\right)$$

Key Concepts

Substitution Method

The substitution method is a common technique used to solve systems of equations. It involves isolating one variable in one of the equations and then substituting that expression into another equation. This approach allows us to solve for one variable at a time, simplifying the process.

• Start with one of the given equations and isolate one of the variables if it's not already isolated.
• Substitute the expression from the isolated variable into the other equation. This effectively reduces the system to a single equation with one variable.
• Solve the new equation to find the value of one variable.
• Use this value to find the other variable by substituting it back into the expression found at the beginning.
• Continue until all variables are solved.

The strength of the substitution method lies in its simplicity, as it transforms a system of equations into easily solvable single-variable equations.

Variables

In mathematics, a variable represents an unknown quantity in an equation or a system of equations. Variables are often denoted by letters such as \(x, y, n, \text{ or } m\). They act as placeholders for values we aim to find through solving equations.
Using variables allows us to model real-world situations mathematically. This includes representing the relationships between different quantities. In our exercise, the variables \(m\) and \(n\) stand for numbers that satisfy both equations of the system simultaneously.
Understanding and properly manipulating variables is crucial in solving mathematical problems. Each variable holds potential answers, making it important to apply proper methods like substitution, to determine their values accurately.

Solving Equations

Solving equations is the heart of algebra and is a crucial skill for solving systems of equations. It involves finding the exact value of variables that satisfy the given equations.
When solving equations, consider these essential steps:

• Identify all expressions and operations in the equation.
• Use algebraic techniques such as addition, subtraction, multiplication, or division to isolate the variable on one side of the equation.
• Perform operations systematically, simplifying the equation gradually to solve for the unknown completely.
• Validate the solution by plugging it back into the original equations to ensure it satisfies all equations.

In our example, solving the equation for \(m\) first and then for \(n\) demonstrates a systematic approach to unraveling the values these variables represent, leading us to the solution of the system.