Solve the System of Equations y=x+z+5 z=-3y-3 2x-y=-4
The problem presents a system of three algebraic equations with three variables, which you are asked to solve. The goal is to find values for the variables x, y, and z that satisfy all three equations simultaneously.
$y = x + z + 5$$z = - 3 y - 3$$2 x - y = - 4$
Answer
Solution:
Step:1
Substitute $x + z + 5$ for $y$ in each equation.
Step:1.1
In the equation $z = -3y - 3$, replace $y$ with $x + z + 5$.
$$z = -3(x + z + 5) - 3$$ $$y = x + z + 5$$ $$2x - y = -4$$
Step:1.2
Carry out the simplification on the right-hand side.
Step:1.2.1
Work on simplifying $-3(x + z + 5) - 3$.
Step:1.2.1.1
Break down the expression term by term.
Step:1.2.1.1.1
Utilize the distributive property.
$$z = -3x - 3z - 15 - 3$$ $$y = x + z + 5$$ $$2x - y = -4$$
Step:1.2.1.1.2
Compute the product of $-3$ and $5$.
$$z = -3x - 3z - 18$$ $$y = x + z + 5$$ $$2x - y = -4$$
Step:1.3
In the equation $2x - y = -4$, substitute $y$ with $x + z + 5$.
$$2x - (x + z + 5) = -4$$ $$z = -3x - 3z - 18$$ $$y = x + z + 5$$
Step:1.4
Simplify the left-hand side.
Step:1.4.1
Focus on simplifying $2x - (x + z + 5)$.
Step:1.4.1.1
Simplify each term individually.
Step:1.4.1.1.1
Apply the distributive property.
$$2x - x - z - 5 = -4$$ $$z = -3x - 3z - 18$$ $$y = x + z + 5$$
Step:1.4.1.1.2
Multiply $-1$ by $5$.
$$x - z - 5 = -4$$ $$z = -3x - 3z - 18$$ $$y = x + z + 5$$
Step:2
Isolate terms without $x$ on the right side of the equation.
Step:2.1
Add $z$ to both sides.
$$x - 5 = -4 + z$$ $$z = -3x - 3z - 18$$ $$y = x + z + 5$$
Step:2.2
Add $5$ to both sides.
$$x = -4 + z + 5$$ $$z = -3x - 3z - 18$$ $$y = x + z + 5$$
Step:2.3
Combine $-4$ and $5$.
$$x = z + 1$$ $$z = -3x - 3z - 18$$ $$y = x + z + 5$$
Step:3
Substitute $z + 1$ for $x$ in each equation.
Step:3.1
In the equation $z = -3x - 3z - 18$, replace $x$ with $z + 1$.
$$z = -3(z + 1) - 3z - 18$$ $$x = z + 1$$ $$y = x + z + 5$$
Step:3.2
Proceed with simplification on the right-hand side.
Step:3.2.1
Work on simplifying $-3(z + 1) - 3z - 18$.
Step:3.2.1.1
Break down the expression term by term.
Step:3.2.1.1.1
Utilize the distributive property.
$$z = -3z - 3 - 3z - 18$$ $$x = z + 1$$ $$y = x + z + 5$$
Step:3.2.1.1.2
Compute the product of $-3$ and $1$.
$$z = -6z - 21$$ $$x = z + 1$$ $$y = x + z + 5$$
Step:3.3
In the equation $y = x + z + 5$, replace $x$ with $z + 1$.
$$y = (z + 1) + z + 5$$ $$z = -6z - 21$$ $$x = z + 1$$
Step:3.4
Simplify the right-hand side.
Step:3.4.1
Focus on simplifying $(z + 1) + z + 5$.
Step:3.4.1.1
Combine $z$ and $z$.
$$y = 2z + 6$$ $$z = -6z - 21$$ $$x = z + 1$$
Step:3.4.1.2
Add $1$ and $5$.
$$y = 2z + 6$$ $$z = -6z - 21$$ $$x = z + 1$$
Step:4
Determine $z$ from $z = -6z - 21$.
Step:4.1
Shift all $z$ terms to the left side.
Step:4.1.1
Add $6z$ to both sides.
$$7z = -21$$ $$y = 2z + 6$$ $$x = z + 1$$
Step:4.2
Divide $7z = -21$ by $7$ to simplify.
Step:4.2.1
Divide $7z$ and $-21$ by $7$.
$$\frac{7z}{7} = \frac{-21}{7}$$ $$y = 2z + 6$$ $$x = z + 1$$
Step:4.2.2
Simplify the left side by canceling the common factor.
Step:4.2.2.1
Cancel out the common factor.
$$z = \frac{-21}{7}$$ $$y = 2z + 6$$ $$x = z + 1$$
Step:4.2.3
Simplify the right side by dividing.
$$z = -3$$ $$y = 2z + 6$$ $$x = z + 1$$
Step:5
Substitute $-3$ for $z$ in the remaining equations.
Step:5.1
In $y = 2z + 6$, replace $z$ with $-3$.
$$y = 2(-3) + 6$$ $$z = -3$$ $$x = z + 1$$
Step:5.2
Simplify the right side.
Step:5.2.1
Compute $2(-3) + 6$.
$$y = 0$$ $$z = -3$$ $$x = z + 1$$
Step:5.3
In $x = z + 1$, substitute $z$ with $-3$.
$$x = (-3) + 1$$ $$y = 0$$ $$z = -3$$
Step:5.4
Simplify the right side.
$$x = -2$$ $$y = 0$$ $$z = -3$$
Step:6
The solution set is the collection of all valid ordered triples.
$(-2, 0, -3)$
Step:7
The solution can be presented in various formats.
Point Form:
$(-2, 0, -3)$
Equation Form:
$x = -2, y = 0, z = -3$
Knowledge Notes:
To solve a system of equations, we often use substitution or elimination methods. The substitution method involves solving one equation for one variable and then substituting that expression into the other equations. This process is repeated until all variables are isolated. The elimination method involves adding or subtracting equations to eliminate one variable, making it easier to solve for the remaining variables.
In the given problem, we used the substitution method. We started by expressing $y$ in terms of $x$ and $z$ from the first equation and then substituted this expression into the other equations. We simplified the equations at each step to isolate variables and solve for them one by one.
The distributive property, which states that $a(b + c) = ab + ac$, is used to expand expressions. This property is essential when simplifying equations in algebra.
When we have an equation like $7z = -21$, we can solve for $z$ by dividing both sides by the coefficient of $z$, which is $7$ in this case. This yields $z = -21 / 7$.
The final step is to substitute the found values back into the original equations to solve for the remaining variables, ensuring that all solutions satisfy the original system of equations. The solution can be expressed as an ordered triple representing the values of $x$, $y$, and $z$ that satisfy all the equations simultaneously.
