Problem 166 - Elementary Algebra

Question

In the following exercises, solve the systems of equations by substitution. $$ \left\\{\begin{array}{l} 2 x+4 y=7 \\ y=-\frac{1}{2} x-4 \end{array}\right. $$

Step-by-Step Solution

Answer

No solution

Step 1 - Substitute Equation

The second equation is already solved for y. Substitute the expression for y from the second equation into the first equation. Start with:\[ y = -\frac{1}{2} x - 4 \]Now, substitute this into the first equation:\[2 x + 4 (-\frac{1}{2} x - 4) = 7 \]

Step 2 - Simplify the Equation

Distribute the 4 into the terms inside the parentheses:\[2 x + 4 \times ( -\frac{1}{2} x ) + 4 \times ( -4 ) = 7 \]\[2 x - 2 x - 16 = 7 \]

Step 3 - Solve for x

Combine like terms on the left side of the equation:\[0 - 16 = 7 \]This simplifies to:\[-16 = 7 \]This is a contradiction, which means there is no solution.

Key Concepts

Systems of equations

A system of equations is a set of two or more equations with the same variables, and the solutions are the points where the equations intersect. For example, consider the given system:
\[ \begin{array}{l} 2x+4y=7 \ y=-\frac{1}{2} x-4 \end{array} \] To solve these systems, you need to find the values of x and y that satisfy both equations simultaneously.

There are different methods for solving systems of equations:

• Graphical method
• Elimination method
• Substitution method

In this exercise, we'll use the substitution method.

Substitution method

The substitution method involves solving one of the equations for one variable and then substituting that expression into the other equation. This reduces the system to a single equation in one variable, which can then be solved.

Let's apply the substitution method to our system: Starting with the second equation, where y is already isolated:
\[ y = -\frac{1}{2}x - 4 \] Substitute this expression for y into the first equation:
\[ 2x + 4 \left( -\frac{1}{2} x - 4 \right) = 7 \] This substitution helps us eliminate y and solve for x.

No solution systems

There are cases where a system of equations has no solution. This happens when the equations describe parallel lines, meaning they never intersect.

Using our system example, simplifying after substitution yields: \[ 2x - 2x - 16 = 7 \] Combining like terms, we get:
\[ 0 - 16 = 7 \] This is a contradiction because -16 cannot equal 7. This contradiction indicates that the system has no solution: the lines are parallel and never meet.

When a system has no solution, it is termed an *inconsistent system.* Understanding this concept helps in identifying parallel lines and preventing unnecessary calculations.