Problem 94 - Algebra 1: Concepts and Skills

Question

Use substitution to solve the system. $$ \begin{aligned} &x+15 y=6\\\ &-x-5 y=84 \end{aligned} $$

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = -129\) and \(y = 9\).

Step 1: Isolate one variable in one of the equations

Starting with the first equation \(x+15y=6\), we isolate x by subtracting 15y from both sides of the equation. This gives us \(x = 6 - 15y\).

Step 2: Substitute for x in the second equation

Now we substitute \(x = 6 - 15y\) from the first equation into the second equation \(-x - 5y = 84\). After substituting, we get \(-(6 - 15y) - 5y = 84\). Simplify and solve for y by distributing the negative sign inside the parentheses and combining like terms, resulting in \(-6 + 15y - 5y = 84\), which simplifies to \(10y = 90\), yielding \(y = 9\).

Step 3: Substitute y into the first equation to solve for x

Now that we have \(y = 9\), we substitute this into the first equation \(x + 15y = 6\). Substituting the value \(y = 9\), we get \(x + 15(9) = 6\), which simplifies to \(x + 135 = 6\). Solving for x we find \(x = 6 - 135 = -129.\)

Key Concepts

Substitution Method

When confronted with a system of linear equations, the substitution method is a powerful tool to find the solution. This method involves three main steps: isolate a variable in one of the equations, substitute the expression for this variable into the other equation, and solve for the second variable, which leaves you with an answer that can be substituted back into any of the original equations to find the remaining variable.

In simple terms, it’s like solving a puzzle where you first find a clue (isolate a variable), then use this clue to solve a riddle (substitute this clue in the other equation), and finally uncover the mystery (find the value of both variables).

The process relies on careful algebraic manipulation and attention to detail to avoid mistakes. It's crucial to perform the same operation to both sides of an equation whenever you're isolating a variable, and to clearly show each step to track your progress effectively.

Systems of Linear Equations

A system of linear equations consists of two or more linear equations with the same set of variables. Imagine these equations as lines on a graph; a solution to the system is a point (or points) where the lines intersect, which means it is the set of numbers that make all equations true simultaneously.

Systems can be solved using a variety of methods, including graphing, substitution, elimination, and matrix operations. The substitution method is one of the most straightforward techniques for systems with fewer variables and when one of the equations can be easily rearranged to isolate one variable.

Algebraic Equations

At the heart of solving systems of equations is understanding algebraic equations. These are mathematical statements indicating that two expressions are equal, containing one or more unknown values known as variables. The challenge and goal of algebra are to determine the value of these variables that make the equation true.

By manipulating these equations through operations such as addition, subtraction, multiplication, division, and factoring, we can isolate the variables and find their values. Clarity and precision in each step ensure we maintain the properties of equality throughout the solving process.

Variable Isolation

Variable isolation is an essential first step in the substitution method. This technique involves rearranging the terms of an equation to express a single variable – the one being isolated – in terms of the other variables and constants present in the equation.

For example, if you have the equation \( a + b = c \), isolating \( a \) would give you \( a = c - b \). In the context of a system of linear equations, isolating a variable allows you to express that variable's value as an algebraic expression, which you can then substitute into another equation, creating a simpler equation that only contains one variable. This simplification is vital in helping us to untangle the often-complex web of variables present in systems of equations.