Problem 82 - Elementary Algebra

Question

$$ \left\\{\begin{array}{l} -3 x-5 y=3 \\ y=\frac{1}{2} x-5 \end{array}\right. $$$$ \left\\{\begin{array}{l} -3 x-5 y=3 \\ y=\frac{1}{2} x-5 \end{array}\right. $$

Step-by-Step Solution

Answer

(4, -3)

Step 1: Substitute the Equation

Use the second equation to express \( y \) in terms of \( x \). Substitute \( y = \frac{1}{2} x - 5 \) into the first equation.

Step 2: Combine Equations

Substitute \( y = \frac{1}{2} x - 5 \) into \( -3x - 5y = 3 \). That gives: \( -3x - 5(\frac{1}{2}x - 5) = 3 \).

Step 3: Simplify the Equation

Simplify the equation to solve for \( x \). This becomes \( -3x - \frac{5}{2}x + 25 = 3 \). Combine the x terms: \( -\frac{11}{2}x + 25 = 3 \).

Step 4: Solve for x

Isolate \( x \) by moving the constant term to the right: \( -\frac{11}{2}x = 3 - 25 \) which simplifies to \( -\frac{11}{2}x = -22 \). Divide both sides by \( -\frac{11}{2} \) to find \( x = 4 \).

Step 5: Find y

Substitute \( x = 4 \) back into the second equation: \( y = \frac{1}{2} \times 4 - 5 \). This simplifies to \( y = 2 - 5 \) which further simplifies to \( y = -3 \).

Step 6: Verify the Solution

Verify the solution \( (x, y) = (4, -3) \) by substituting both \( x \) and \( y \) back into the original equations to ensure both are satisfied.

Key Concepts

Simultaneous Equations

Simultaneous equations are a set of equations with multiple variables.
The goal is to find a single solution that satisfies all equations in the system.
In our case, we had two equations involving the variables \(x\) and \(y\):

• \(-3x - 5y = 3\)
• \(y = \frac{1}{2}x - 5\)

Solving such problems involves finding the values of \(x\) and \(y\) that make both equations true simultaneously.

Substitution Method

The substitution method is a very useful technique for solving simultaneous equations.
It involves solving one equation for one variable and then substituting this expression into the other equation.
In our example, the second equation \(y = \frac{1}{2}x - 5\) is already solved for \(y\).
We substitute this expression into the first equation \(-3x - 5y = 3\) as follows:

• \(-3x - 5(\frac{1}{2}x - 5) = 3\)

This substitution transforms the problem into a single-variable equation, making it easier to solve.

Solving for Variables

Once we substitute, we need to simplify and solve for the variable.
Following the substitution, we get:

• \(-3x - 5(\frac{1}{2}x - 5) = 3\)

Simplifying step-by-step, we have:

• \(-3x - \frac{5}{2}x + 25 = 3\)
• \(-\frac{11}{2}x + 25 = 3\)
• \(-\frac{11}{2}x = 3 - 25\)
• \(-\frac{11}{2}x = -22\)
Finally, solving for \(x\) yields:

• \(x = 4\)
We then substitute \(x = 4\) back into the second equation \(y = \frac{1}{2}x - 5\):

• \(y = \frac{1}{2} \times 4 - 5\)
• \(y = 2 - 5\)
• \(y = -3\)

Verification of Solutions

After finding the values of \(x\) and \(y\), it's crucial to verify the solution.
This ensures that our solution satisfies both original equations.
Substituting \(x = 4\) and \(y = -3\) back into the first equation:

• \(-3(4) - 5(-3) = 3\)
• \(-12 + 15 = 3\)
• \(3 = 3\)

Next, verify in the second equation:

• \(y = \frac{1}{2}(4) - 5\)
• \(-3 = 2 - 5\)
• \(-3 = -3\)

Both equations are satisfied, confirming the solution \((x, y) = (4, -3)\) is correct.