Problem 49 - Understanding Elementary Algebra with Geometry

Question

In Exercises \(29-62,\) solve the system of equations using any method you choose. $$\left\\{\begin{array}{l} 7 x+4 y=5 \\ 4 x+3 y=0 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution is \( x = 3 \) and \( y = -4 \).

Step 1: Write the system of equations

Given the system of equations: \[ \begin{cases} 7x + 4y = 5 \ 4x + 3y = 0 \end{cases} \]

Step 2: Choose a method to solve the system

In this example, we will use the substitution method.

Step 3: Solve one of the equations for one variable

From the second equation, solve for \( y \): \[ 4x + 3y = 0 \implies 3y = -4x \implies y = -\frac{4}{3}x \]

Step 4: Substitute the expression into the other equation

Substitute \( y = -\frac{4}{3}x \) into the first equation: \[ 7x + 4\left( -\frac{4}{3}x \right) = 5 \]

Step 5: Simplify and solve for \( x \)

Simplify the equation: \[ 7x - \frac{16}{3}x = 5 \] Combine like terms: \[ \frac{21x - 16x}{3} = 5 \] \[ \frac{5x}{3} = 5 \] Multiply both sides by 3: \[ 5x = 15 \] Divide by 5: \[ x = 3 \]

Step 6: Substitute \( x \) back into the expression for \( y \)

Substitute \( x = 3 \) back into \( y = -\frac{4}{3}x \): \[ y = -\frac{4}{3} \times 3 \] \[ y = -4 \]

Step 7: Write the solution

The solution to the system of equations is \( x = 3 \) and \( y = -4 \).

Key Concepts

substitution method

The substitution method is a technique for solving a system of equations by expressing one variable in terms of another. In this method, one equation is solved for one variable, and that expression is then substituted into the other equation. This allows us to solve for the remaining variable easily.

In the given exercise, the second equation was used to solve for y in terms of x:
4x + 3y = 0 ⇒ y = -\(\frac{4}{3}\times x\)
This expression for y was then substituted into the first equation to solve for x. Once x was found, it was substituted back to solve for y. This sequential substitution simplifies the process of finding the solution.

While this method is straightforward and effective for many systems, it is particularly useful when one equation is easily solvable for one variable. Remember to always substitute back to check your solutions in the original equations!

linear equations

Linear equations are equations of the first degree, meaning they involve only the first powers of the variables. The solution to a system of linear equations is the point(s) where the equations intersect on a graph.

In our exercise, we dealt with two such linear equations:
7x + 4y = 5
4x + 3y = 0
These represent straight lines when plotted on a coordinate plane. The solution, (x=3, y=-4), is the point where these two lines intersect.

Linear systems can be solved using various methods, including substitution, elimination, and graphing. It’s important to recognize that a system of linear equations may have:

• One solution (intersecting lines)
• No solution (parallel lines)
• Infinitely many solutions (coinciding lines)

Understanding the nature of linear equations helps us visualize and solve these systems effectively.

algebraic solution steps

The algebraic solution steps refer to the organized approach taken to solve equations systematically. For our system of equations, the following steps were meticulously followed:

• **Step 1:** Identify the given system of equations.
• **Step 2:** Choose a method for solving the system (substitution in our case).
• **Step 3:** Solve one of the equations for one variable (solved 4x + 3y = 0 for y).
• **Step 4:** Substitute this expression in the other equation (substitute y in 7x + 4y = 5).
• **Step 5:** Simplify and solve for the other variable (solved for x).
• **Step 6:** Substitute back to find the remaining variable (substitute x back to find y).
• **Step 7:** Write down the final solution (x = 3, y = -4).

Breaking down the problem into clear, logical steps not only makes the solution accessible but also helps in better understanding and retention. Always remember to substitute your solution back into the original equations to verify its correctness!