Problem 158 - Elementary Algebra

Question

In the following exercises, solve the systems of equations by substitution. $$ \left\\{\begin{array}{l} x-4 y=-1 \\ -3 x+12 y=3 \end{array}\right. $$

Step-by-Step Solution

Answer

The system has infinitely many solutions.

Step 1: Solve one of the equations for one variable

First, isolate one of the variables in one of the equations. Let's solve the first equation for the variable x: x - 4y = -1 Add 4y to both sides: x = 4y - 1

Step 2: Substitute the expression into the other equation

Next, substitute the expression for x from the first equation into the second equation: -3x + 12y = 3 Substitute x = 4y - 1: -3(4y - 1) + 12y = 3

Step 3: Simplify and solve for y

Simplify the equation and solve for y: -12y + 3 + 12y = 3 Combine like terms: 3 = 3 Since this statement is always true, it means that there are infinitely many solutions that lie on the same line. Therefore, the system of equations has infinitely many solutions.

Key Concepts

substitution method

The substitution method is a powerful technique used to solve systems of equations by expressing one variable in terms of another. This allows you to replace one variable in the second equation, simplifying the problem to a single-variable equation.
To start, you solve one of the equations for one of the variables. In the given problem, we solved for `x` in terms of `y` from the first equation:
x - 4y = -1
Adding 4y to both sides, we get:
x = 4y - 1.
This new expression for `x` is then substituted into the other equation. This approach reduces the second equation to a single variable equation, making it easier to solve.
By changing the problem into one that is simpler and more straightforward, the substitution method provides a step-by-step pathway towards the solution of the system.

solving equations

Solving equations involves finding the values of the variables that make all equations true simultaneously.
After substituting x = 4y - 1 into the second equation -3x + 12y = 3, we get:
-3(4y - 1) + 12y = 3.
Simplifying this, we get:
-12y + 3 + 12y = 3.
When we combine like terms, we notice that both y terms cancel each other out leading to the equation:
3 = 3.
This simplified form shows that the equation holds true for any value of y.
It implies that the original system of equations does not limit y to any particular value. In turn, this means there are multiple values for y that make the system true.

infinitely many solutions

When a system of equations has infinitely many solutions, it means that the equations represent the same line or are dependent.
In this case, every point on the line is a solution to both equations.
As seen in the equation 3 = 3, which is always true, it indicates that for every `y` substitute, you will get a corresponding `x` that satisfies both equations.
Therefore, this is a typical scenario where the lines are identical and thus every point on the line is a solution.
In conclusion, the outcome of having infinitely many solutions shows that the system of equations doesn't just intersect at a single point but along an entire line. This is crucial in understanding dependencies within systems of equations.