Solve by Substitution y=-1x-5 4x-8y=4
This problem involves solving a system of two linear equations using the substitution method. The first equation is y = -1x - 5, and the second one is 4x - 8y = 4. To apply the substitution method, you would use the expression given for y in the first equation and substitute it into the second equation in place of y. The goal is to solve for x first and then use the value of x to find y.
$y = - 1 x - 5$$4 x - 8 y = 4$
Answer
Solution:
Step 1:
Express $-1x$ as $-x$. We have $y = -x - 5$ and $4x - 8y = 4$.
Step 2:
Substitute $-x - 5$ for $y$ in each equation.
Step 2.1:
In the equation $4x - 8y = 4$, replace $y$ with $-x - 5$ to get $4x - 8(-x - 5) = 4$. Keep $y = -x - 5$ for later use.
Step 2.2:
Begin simplifying the equation.
Step 2.2.1:
Expand $4x - 8(-x - 5)$.
Step 2.2.1.1:
Distribute $-8$ across $-x$ and $-5$ to get $4x + 8x + 40 = 4$. Retain $y = -x - 5$.
Step 2.2.1.2:
Combine like terms $4x$ and $8x$ to obtain $12x + 40 = 4$. Keep $y = -x - 5$.
Step 3:
Isolate $x$ in the equation $12x + 40 = 4$.
Step 3.1:
Shift terms not containing $x$ to the opposite side.
Step 3.1.1:
Subtract $40$ from both sides to get $12x = 4 - 40$. Keep $y = -x - 5$.
Step 3.1.2:
Compute $4 - 40$ to find $12x = -36$. Retain $y = -x - 5$.
Step 3.2:
Divide the equation $12x = -36$ by $12$ to solve for $x$.
Step 3.2.1:
Divide $12x$ and $-36$ by $12$ to get $\frac{12x}{12} = \frac{-36}{12}$. Keep $y = -x - 5$.
Step 3.2.2:
Cancel out the common factor of $12$.
Step 3.2.2.1:
Eliminate the common factor to simplify to $x = \frac{-36}{12}$. Retain $y = -x - 5$.
Step 3.2.2.2:
Reduce $x$ to $x = -3$. Keep $y = -x - 5$.
Step 4:
Insert $-3$ for $x$ in the remaining equations.
Step 4.1:
In the equation $y = -x - 5$, substitute $x$ with $-3$ to get $y = -(-3) - 5$. Keep $x = -3$.
Step 4.2:
Simplify the equation.
Step 4.2.1:
Resolve $-(-3) - 5$.
Step 4.2.1.1:
Multiply $-1$ by $-3$ to get $y = 3 - 5$. Retain $x = -3$.
Step 4.2.1.2:
Subtract $5$ from $3$ to find $y = -2$. Keep $x = -3$.
Step 5:
The solution set consists of the ordered pair that satisfies both equations: $(-3, -2)$.
Step 6:
The solution can be presented in various formats.
- Point Form: $(-3, -2)$
- Equation Form: $x = -3, y = -2$
Step 7:
(No further steps required)
Knowledge Notes:
Substitution Method: A technique used to solve systems of equations where one equation is solved for one variable in terms of the others, and then substituted into the remaining equations.
Simplification: The process of reducing expressions to their simplest form, often by combining like terms and applying arithmetic operations.
Distributive Property: A property that allows us to multiply a sum by a number by multiplying each addend separately and then sum the products: $a(b + c) = ab + ac$.
Isolating Variables: The process of manipulating an equation to get a variable by itself on one side of the equation, often to solve for that variable.
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 satisfies all equations simultaneously.
Ordered Pair: A pair of numbers used to locate a point on a coordinate plane, typically written in the form $(x, y)$.
Latex Format: A typesetting system that is widely used for mathematical and scientific documents, due to its powerful handling of formulas and bibliographies.
