Question

Verify that the values of the variables listed are solutions of the system of equations. $$ \begin{array}{l} \left\\{\begin{array}{l} 3 x-4 y=4 \\ \frac{1}{2} x-3 y=-\frac{1}{2} \end{array}\right. \\ x=2, y=\frac{1}{2} ;\left(2, \frac{1}{2}\right) \end{array} $$

Short answer

Yes, \((2, \frac{1}{2})\) is a solution.

Step-by-step solution

Substitute the values into the first equation

Given \( x = 2 \) and \( y = \frac{1}{2} \), substitute these values into the first equation \( 3x - 4y = 4 \). The equation becomes: \[ 3(2) - 4\left( \frac{1}{2} \right) = 4 \].

Simplify the first equation

Simplify the left side of the equation: \[ 6 - 2 = 4 \]. The equation simplifies to \( 4 = 4 \), which is true.

Substitute the values into the second equation

Next, substitute \( x = 2 \) and \( y = \frac{1}{2} \) into the second equation \( \frac{1}{2}x - 3y = -\frac{1}{2} \). The equation becomes: \[ \frac{1}{2}(2) - 3\left( \frac{1}{2} \right) = -\frac{1}{2} \].

Simplify the second equation

Simplify the left side of the equation: \[ 1 - \frac{3}{2} = -\frac{1}{2} \]. The equation simplifies to \( -\frac{1}{2} = -\frac{1}{2} \), which is also true.

Conclusion

Since both equations are true when \( x = 2 \) and \( y = \frac{1}{2} \), these values are solutions to the system of equations.

Key concepts

Substitution Method

The substitution method is a technique used to solve systems of equations. It's especially handy when one equation is already in terms of one variable. Here's how it works: you solve one of the equations for one variable, then substitute that expression into the other equation.
In our example, given the system of equations:
\[ \begin{array}{l} 3x - 4y = 4 \ \frac{1}{2}x - 3y = -\frac{1}{2} \end{array} \]
Instead of explicitly solving one equation for a variable first, we were directly given the values of the variables (\ x=2 \ and \ y=\frac{1}{2} ). We verified these values by substituting them into both equations.
This convenient method helps simplify complex systems and can be a quick way to find solutions when the conditions are right.

Verification of Solutions

Verification of solutions ensures that the proposed values satisfy both equations in a system. Here’s the step-by-step process:

• Substitute the given values into each equation.
• Simplify both sides of the equations to see if they are equal.

For the given system:

• First equation: \[ 3x - 4y = 4 \]
• Second equation: \[ \frac{1}{2}x - 3y = -\frac{1}{2} \]

With \( x=2 \) and \( y=\frac{1}{2} \):

• Substituted the values in the first equation:
  \[ 3(2) - 4\left( \frac{1}{2} \right) = 4 \]
  After simplifying:
  \[ 6 - 2 = 4 \], so \[ 4 = 4 \] which is true.
• Substituted the values in the second equation:
  \[ \frac{1}{2}(2) - 3\left( \frac{1}{2} \right) = -\frac{1}{2} \]
  After simplifying:
  \[ 1 - \frac{3}{2} = -\frac{1}{2} \], so \[ -\frac{1}{2} = -\frac{1}{2} \] which is also true.

Both equations are satisfied by the provided solutions, verifying their correctness.

Linear Equations

Linear equations are equations of the first degree, meaning they involve only the first powers of variables. These equations graph as straight lines in a coordinate plane, and they form the foundation of algebraic problem-solving.
In our system of equations:
\[ \begin{array}{l} 3x - 4y = 4 \ \frac{1}{2}x - 3y = -\frac{1}{2} \end{array} \], both are linear equations.
Each linear equation represents a straight line, and the solution to the system is the point where these two lines intersect. The given solution \ (x=2, y=\frac{1}{2}) \ lies at the intersection point of these lines.
Understanding linear equations is vital because they are used in various fields such as physics, economics, and engineering to model relationships between quantities. Mastering these equations helps build a strong foundation for more advanced mathematical concepts.