Problem 26 - Algebra A Combined Function

Question

Solve each system of equations by the substitution method. $$ \left\\{\begin{array}{l} 2 x-y=-7 \\ 4 x-3 y=-11 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \((-5, -3)\).

Step 1: Solve the First Equation for y

We start with the first equation from the system: \[ 2x - y = -7 \]Rearrange this equation to solve for \(y\):\[ y = 2x + 7 \]This expression for \(y\) will be used to substitute into the second equation.

Step 2: Substitute the Expression for y into the Second Equation

Take the expression found for \(y\):\[ y = 2x + 7 \]and substitute it into the second equation:\[ 4x - 3y = -11 \]Replace \(y\) with \(2x + 7\):\[ 4x - 3(2x + 7) = -11 \]

Step 3: Solve for x

Distribute and simplify the equation:\[ 4x - 6x - 21 = -11 \]Combine like terms:\[ -2x - 21 = -11 \]Add 21 to both sides to isolate the term involving \(x\):\[ -2x = 10 \]Finally, divide by \(-2\) to solve for \(x\):\[ x = -5 \]

Step 4: Substitute x back into the Expression for y

Now that we have \(x = -5\), substitute this value into the expression for \(y\) from Step 1:\[ y = 2(-5) + 7 \]Simplify:\[ y = -10 + 7 = -3 \]

Step 5: State the Solution to the System

The values \(x = -5\) and \(y = -3\) solve the given system of equations. The solution to the system is:\((-5, -3)\).

Key Concepts

System of Equations

A system of equations is a set of two or more equations that share the same variables. These variables are usually represented by symbols like \(x\) and \(y\). The goal is to find a common solution that satisfies all equations in the system.
In this exercise, we have the following system of linear equations:

• \(2x - y = -7\)
• \(4x - 3y = -11\)

To solve this system, we use the substitution method, which involves isolating one variable in terms of the others and substituting this expression into the other equation.
This approach helps determine the values of the variables that meet all conditions provided by the equations in the system.

Solving Equations

Solving equations involves finding the values of variables that make the equations true. In the context of the substitution method, we start by manipulating one equation to express one variable explicitly via another variable.
Let's look at how we did this process:

• First, from \(2x-y=-7\), we solved for \(y\) by rearranging it to \(y=2x+7\).
• Next, we substituted \(y=2x+7\) into the second equation \(4x-3y=-11\). This replacement gives us one equation with just one variable, simplifying the problem.

By substituting and simplifying, we translate the original system of equations into a solvable format.
This simplification is crucial because it allows us to solve for one variable at a time, making complex problems more manageable.

Linear Equations

Linear equations describe relationships where the variable expressions are first degree, meaning they involve no exponents higher than one. These equations graph as straight lines on a coordinate plane.
In our system, both equations are linear:

• \(2x-y=-7\)
• \(4x-3y=-11\)

Each equation forms a line, and the solution to the system is the point where these lines intersect.
This solution can often be visualized on a graph, but here, algebraic approaches like substitution make it easy to find the precise intersection point, which in this exercise is \((-5, -3)\).
Understanding linear relationships is essential, as they form the basis for more complex math topics and real-world modeling.