Problem 16 - Algebra for College Students

Question

Solve each system by using the substitution method. \(\left(\begin{array}{l}y=\frac{3}{4} x+5 \\ 4 x-3 y=-1\end{array}\right)\)

Step-by-Step Solution

Answer

The solution is \( (8, 11) \).

Step 1: Identify the equations

The system of equations given is: 1. \( y = \frac{3}{4}x + 5 \)2. \( 4x - 3y = -1 \)

Step 2: Substitute for y

Since the first equation is already solved for \( y \), substitute \( y = \frac{3}{4}x + 5 \) into the second equation. This gives us:\[4x - 3\left(\frac{3}{4}x + 5\right) = -1\]

Step 3: Distribute and simplify

Distribute the \( -3 \) across the terms inside the parentheses:\[4x - \frac{9}{4}x - 15 = -1\]Combine like terms:\[\frac{16}{4}x - \frac{9}{4}x - 15 = -1\]\[\frac{7}{4}x - 15 = -1\]

Step 4: Solve for x

Add 15 to both sides to isolate the \( x \) term:\[\frac{7}{4}x = 14\]Multiply both sides by \( \frac{4}{7} \) to solve for \( x \):\[x = \frac{4}{7} \times 14\]\[x = 8\]

Step 5: Substitute x back into the first equation

Substitute \( x = 8 \) back into the first equation to solve for \( y \):\[y = \frac{3}{4} \times 8 + 5\]\[y = 6 + 5\]\[y = 11\]

Step 6: Write the solution as an ordered pair

The solution to the system of equations is \( (x, y) = (8, 11) \).

Key Concepts

Systems of Equations

A system of equations consists of two or more equations that share the same variables. In simpler terms, it’s a set of equations you solve together since they describe relationships between the same variables. For example, in our problem, we have the system:

• Equation 1: \( y = \frac{3}{4}x + 5 \)
• Equation 2: \( 4x - 3y = -1 \)

Solving systems of equations is essential in algebra because it helps us find the values of the variables that satisfy all equations at once. Imagine plotting both equations on a graph — the solution is where the lines intersect. This intersection point represents the values of \( x \) and \( y \) that work for both equations.

Linear Equations

Linear equations are equations in which the variables are only raised to the power of one. They take the form of a straight line when graphed on a coordinate plane. The standard form of a linear equation is \( Ax + By = C \) where A, B, and C are constants. In our exercise, both equations are linear:

• In the first equation, \( y = \frac{3}{4}x + 5 \), \( y \) is expressed in terms of \( x \).
• In the second equation, \( 4x - 3y = -1 \), both variables \( x \) and \( y \) are on the same side initially.

Linear equations showcase relationships where the change in one variable is proportional to the change in another. Understanding them is crucial since many real-world phenomena can be modeled using linear relationships.

Algebraic Substitution

Algebraic substitution is a method used to solve systems of equations where we solve one equation for one variable and then substitute that value into another equation. It allows us to reduce the number of variables in an equation, making it simpler to solve.In our given problem, the substitution method started by recognizing equation 1 as already solved for \( y \). This enabled us to express \( y \) in terms of \( x \) and substitute it into equation 2. The process involved:

• Replacing \( y \) in Equation 2 with \( \frac{3}{4}x + 5 \).
• Simplifying to remove fractions and combine like terms.

By substituting and simplifying, we transformed a two-variable equation into a single-variable equation, which is much easier to tackle.

Problem Solving in Algebra

Problem-solving in algebra often involves a series of steps to find an unknown. We frequently need to determine how different variables relate within the constraints given by equations. Applying methods like substitution is crucial when equations are intertwined.

• Firstly, we identified an equation that was easy to manipulate for substitution.
• Then, we substituted and simplified until we could solve for one variable.
• Finally, we solved for the remaining variable using the known value.

Each step builds upon the previous one, leading us closer to the solution. This systematic approach is not only useful in mathematical problems but also enhances logical reasoning skills.