Problem 23 - Algebra for College Students

Question

Solve each system by using the substitution method. \(\left(\begin{array}{l}4 x+3 y=-7 \\ 3 x-2 y=16\end{array}\right)\)

Step-by-Step Solution

Answer

The solution is \(x = 2\) and \(y = -5\).

Step 1: Solve One Equation for a Variable

Begin by solving the first equation for one of the variables. Let's solve for \(y\) in terms of \(x\) in the first equation: \[4x + 3y = -7\]Subtract \(4x\) from both sides:\[3y = -7 - 4x\]Now, divide by 3:\[y = \frac{-7 - 4x}{3}\]

Step 2: Substitute into the Other Equation

Take the expression we found for \(y\) and substitute it into the second original equation:\[3x - 2y = 16\]Replace \(y\) with \(\frac{-7 - 4x}{3}\):\[3x - 2\left(\frac{-7 - 4x}{3}\right) = 16\]Simplify this equation.

Step 3: Simplify and Solve for x

Distribute the \(-2\) across the fraction:\[3x + \frac{14 + 8x}{3} = 16\]Multiply everything by 3 to eliminate the fraction:\[9x + 14 + 8x = 48\]Combine like terms:\[17x + 14 = 48\]Subtract 14 from both sides:\[17x = 34\]Divide by 17 to solve for \(x\):\[x = 2\]

Step 4: Substitute Back to Find y

Use the value of \(x\) to find \(y\). Substitute \(x = 2\) back into \(y = \frac{-7 - 4x}{3}\):\[y = \frac{-7 - 4(2)}{3}\]Calculate:\[y = \frac{-7 - 8}{3} = \frac{-15}{3} = -5\]

Step 5: State the Solution

The solution to the system of equations is \(x = 2\) and \(y = -5\). Thus, the point of intersection for the two lines represented by these equations is \((2, -5)\).

Key Concepts

System of Equations

A system of equations refers to a collection of two or more equations with a common set of variables. In the given exercise, the system consists of two equations:

• 4x + 3y = -7
• 3x - 2y = 16

These linear equations involve two variables, x and y. The solution to a system of equations is the set of values for these variables that make all the equations true simultaneously. For instance, the values x = 2 and y = -5 satisfy both of these equations, representing the point of intersection of their corresponding lines on a graph. Solving a system of equations is an essential skill in algebra, as it allows us to find the precise values that balance all equations involved.

Linear Equations

Linear equations are equations in which each term is either a constant or the product of a constant and a single variable. In other words, the highest exponent of the variable is 1. They are called 'linear' because their graphical representation is a straight line. This particular system of equations involves two linear equations:

• 4x + 3y = -7
• 3x - 2y = 16

These equations fall into the category of linear equations because variables x and y are both raised to the power of 1. Solving such equations involves finding the point where these lines intersect, if they do at all. Real-world phenomena often involve linear relationships, making understanding these equations crucial.

Algebraic Solution

The algebraic solution of a system of equations refers to solving the equations using algebraic methods. There are several approaches, and one common method is substitution, particularly suitable for equations in two variables. In this approach, we first solve one of the equations for one variable in terms of the other variable. For our exercise, we solved the first equation for y:

• y = \(\frac{-7 - 4x}{3}\)

Next, we substitute this expression for y in the second equation and solve for x. Once we find the value for x, we substitute it back into the expression for y. This step-by-step substitution method is straightforward and effective, especially when tackling two-variable systems.

Equations in Two Variables

Equations in two variables involve expressions with two distinct variables, often x and y. Each equation provides a relationship between these two variables. In our exercise, the two equations are:

• 4x + 3y = -7
• 3x - 2y = 16

The goal is to find the values of x and y that satisfy both equations simultaneously. This involves finding a common solution, where the pair \((x, y)\) is valid for both lines represented by these equations. In graphs, each equation corresponds to a line, and the solution represents the intersection point of these lines, illustrating where both equations hold true together. Understanding and solving such equations are foundational aspects of algebra and are widely applicable in various fields of science and engineering.