Problem 4 - Contemporary Precalculus

Question

Determine whether the given values of \(x, y,\) and z are a solution of the system of equations. $$\begin{array}{r}x=.3, y=.7 \\\4 x-1.2 y=.36 \\\3.1 x+2 y=4.7\end{array}$$

Step-by-Step Solution

Answer

Answer: No, the given values for x and y do not satisfy the system of equations.

Step 1: Substitute the given values into the first equation

Replace x with 0.3 and y with 0.7 in the first equation: $$4x - 1.2y = 0.36$$ $$4(0.3) - 1.2(0.7)$$ Now we will calculate the result.

Step 2: Calculate the result of the first equation

Perform the multiplication and subtraction in the equation: $$1.2 - 0.84 = 0.36$$ Since the equation is true, we will proceed with substituting the values into the second equation.

Step 3: Substitute the given values into the second equation

Replace x with 0.3 and y with 0.7 in the second equation: $$3.1x + 2y = 4.7$$ $$3.1(0.3) + 2(0.7)$$ Now we will calculate the result.

Step 4: Calculate the result of the second equation

Perform the multiplication and addition in the equation: $$0.93 + 1.4 = 2.33$$ Since the equation 3.1x + 2y = 4.7 resulted in 2.33 = 4.7, which is not true, we can conclude that the given values for x and y are not a solution for the system of equations.

Key Concepts

Substitution Method

The substitution method is a technique used to solve systems of linear equations. Here, you replace one variable with an expression obtained from another equation that involves the second variable. This way, you simplify the system to a single equation with one unknown, making it more manageable to solve.

• You first need to express one variable in terms of the other using one of the equations.
• Then, substitute this expression into the other equation to find the value of one variable.
• Finally, use the found value to determine the second variable using the initial expression you derived.

This method is particularly useful when one variable is easily isolated. It helps simplify solving a system of equations, as seen in our exercise where specific values were substituted in place of the variables.

Checking Solutions

When you're given potential solutions for a system of equations, verifying them is crucial. Checking involves substituting these values into each equation of the system to ensure they hold true.
Replace each variable in the equations with the given numerical values. Perform the arithmetic operations following the order of operations. If both sides of the equation equal for all equations in the system, the given values are indeed a solution.
In our exercise, substituting the values in both equations revealed:

• The first equation was valid since both sides equaled.
• The second equation was not, as the sides did not match.

This verifies that the proposed values do not solve the entire system correctly.

Linear Equations

Linear equations involve variables with exponents of one, forming straight lines when graphed. A system of linear equations consists of two or more equations working together, each with their own intercepts and slopes on a graph.
In the given exercise, the system contains two linear equations:

• 4x - 1.2y = 0.36
• 3.1x + 2y = 4.7

Each equation presents a linear relationship between the variables. By solving these equations simultaneously, we are looking for common values of the variables that satisfy both equations, representing their intersection point on a graph.
Understanding linear equations is fundamental in determining the solutions of many mathematical and real-world problems, as they describe relationships between quantities.