Solve the System of Equations x-3y=6 -8x-y=6
The given problem is asking to find the values of the variables x and y that simultaneously satisfy two linear equations. These equations are provided in the form of algebraic expressions with each equation representing a straight line in a two-dimensional Cartesian plane. The solution to the system of equations will be the point (x, y) where these two lines intersect. To solve the system, various methods can be used such as substitution, elimination, or graphing, but the specific method to be employed is not stated in the question. The goal is to manipulate these equations in such a way that the values of x and y are determined.
$x - 3 y = 6$$- 8 x - y = 6$
Answer
Solution:
Step:1
Isolate $x$ in the first equation by adding $3y$ to both sides: $x = 6 + 3y$. Keep the second equation as is: $-8x - y = 6$.
Step:2
Substitute the expression for $x$ from the first equation into the second equation.
Step:2.1
In the equation $-8x - y = 6$, replace $x$ with $6 + 3y$: $-8(6 + 3y) - y = 6$.
Step:2.2
Expand and simplify the equation.
Step:2.2.1
Distribute $-8$ across $(6 + 3y)$: $-8 \cdot 6 - 8 \cdot 3y - y = 6$.
Step:2.2.2
Perform the multiplication: $-48 - 24y - y = 6$.
Step:2.2.3
Combine like terms: $-48 - 25y = 6$.
Step:3
Solve for $y$ in the equation $-48 - 25y = 6$.
Step:3.1
Isolate the $y$ term by adding $48$ to both sides: $-25y = 6 + 48$.
Step:3.2
Simplify and solve for $y$.
Step:3.2.1
Combine the constants on the right side: $-25y = 54$.
Step:3.2.2
Divide both sides by $-25$: $y = \frac{54}{-25}$.
Step:3.2.3
Simplify the fraction: $y = -\frac{54}{25}$.
Step:4
Substitute the value of $y$ back into the first equation to solve for $x$.
Step:4.1
In the equation $x = 6 + 3y$, replace $y$ with $-\frac{54}{25}$: $x = 6 + 3\left(-\frac{54}{25}\right)$.
Step:4.2
Simplify the expression to find $x$.
Step:4.2.1
Multiply $3$ by $-\frac{54}{25}$: $x = 6 - \frac{162}{25}$.
Step:4.2.2
Convert $6$ to a fraction with a denominator of $25$: $x = \frac{6 \cdot 25}{25} - \frac{162}{25}$.
Step:4.2.3
Combine the fractions: $x = \frac{150 - 162}{25}$.
Step:4.2.4
Simplify the numerator: $x = \frac{-12}{25}$.
Step:5
The solution set for the system of equations is the pair of values for $x$ and $y$.
Step:6
The solution can be expressed in different formats.
Point Form:
$(x, y) = \left(-\frac{12}{25}, -\frac{54}{25}\right)$
Equation Form:
$x = -\frac{12}{25}, y = -\frac{54}{25}$
Knowledge Notes:
Substitution Method: This method involves solving one of the equations for one variable and substituting the result into the other equation.
Isolation of Variables: To solve for one variable in terms of another, we isolate the variable on one side of the equation.
Distributive Property: This property is used to multiply a single term and two or more terms inside a set of parentheses.
Combining Like Terms: This involves adding or subtracting terms that have the same variable raised to the same power.
Solving Linear Equations: The process of finding the value of the variable that makes the equation true.
Fraction Simplification: Reducing fractions to their simplest form by dividing the numerator and denominator by their greatest common factor.
System of Equations: A set of two or more equations with the same variables. The solution to the system is the set of values that satisfy all equations simultaneously.
