Problem 66 - Algebra 2

Question

Find the value of \(a\) that makes each system a dependent system. \(\left\\{\begin{array}{l}{y=\frac{x}{2}+4} \\ {2 y-x=a}\end{array}\right.\)

Step-by-Step Solution

Answer

The value of \(a\) that makes the system a dependent system is \(a = 8\).

Step 1: Express x from the first equation

Start with the first equation of the system, which is in slope-intercept form: \(y = \frac{x}{2} + 4\). This can be rearranged to express \(x\) in terms of \(y\) by multiplying both sides of the equation by 2: \(x = 2y - 8\).

Step 2: Plug the expression for x into the second equation

Take the expression for \(x\) from step 1 and substitute it into the second equation of the system: \(2y - (2y - 8) = a\).

Step 3: Simplify the resulting equation

Simplify the equation by distributing and combining like terms. This yields \(2y - 2y + 8 = a\), which simplifies further to \(a = 8\).

Step 4: Analyze the system for dependence

With \(a = 8\), the original second equation becomes \(2y - x = 8\), which is equivalent to the expression we found for \(x\) when multiplied by -1: \(-x = -2y + 8\). This means the two equations are equivalent, therefore the system is dependent.

Key Concepts

System of Linear Equations

Understanding a system of linear equations is fundamental in algebra. Such a system comprises two or more linear equations with the same set of variables.

A system of equations can have one of three possible relationships: independent, dependent, or inconsistent. An independent system has exactly one solution pair, representing the point where the lines intersect. In a dependent system, the equations describe the same line, meaning there are infinitely many solutions since any point on the line satisfies both equations. An inconsistent system has no solutions, as the lines are parallel and never intersect.

The exercise you encountered deals with a dependent system. You are given two equations and need to determine the value of the variable 'a' so that the system becomes dependent. The concept of dependence in this context means the two lines are one and the same, so they coincide completely.

Slope-Intercept Form

The slope-intercept form is an equation of a line defined as \(y = mx + b\), where \(m\) stands for the slope of the line and \(b\) gives the y-intercept, the point where the line crosses the y-axis.

From the first given equation, \(y = \frac{x}{2} + 4\), we identify the slope as \(\frac{1}{2}\) and the y-intercept as 4. This means for every one unit increase in \(x\), \(y\) increases by half a unit. The slope-intercept form is useful because it directly shows the rate at which \(y\) changes with \(x\) and the starting point on the y-axis.

By converting one or both equations to slope-intercept form during problem-solving, you can compare their slopes and y-intercepts, which is a quick method to determine if you’re dealing with independent, dependent, or inconsistent systems.

Substitution Method

The substitution method is one of the techniques used to solve systems of linear equations. This method involves isolating one variable in one equation, then substituting the result into another equation.

In the case of the exercise, the first step is to express \(x\) in terms of \(y\), resulting in \(x = 2y - 8\). Then, this expression is substituted into the second equation where \(x\) appears. This allows you to find the value of the variable 'a' that makes the system dependent.

The reason substitution works well here is because it simplifies the system to a single equation with one variable, which can be solved directly. For the system to be dependent, the substitution must simplify such that the variable \(y\) cancels out, leaving an equation involving only 'a'. As seen in the solution steps, after substitution and simplification, we find that when \(a = 8\), the equations are essentially the same. This is when the system's dependence is fulfilled.