Solve the System of Equations 9y=-17x+72 -72+17x=-9y
The given problem is a system of two linear equations with two variables, x and y. The question asks to find the values of x and y that satisfy both equations simultaneously. The equations are presented in a slightly different form each time, but they represent the same relationship between x and y, and thus, the system must be solved to find the point of intersection where both equations are true at the same time.
$9 y = - 17 x + 72$$- 72 + 17 x = - 9 y$
Answer
Solution:
Step:1
Isolate $y$ in the equation $9y = -17x + 72$ by dividing every term by $9$.
Step:1.1
Divide each term in $9y = -17x + 72$ by $9$ to get $\frac{9y}{9} = \frac{-17x}{9} + \frac{72}{9}$ and rewrite $-72 + 17x = -9y$ for clarity.
Step:1.2
Simplify the equation involving $y$.
Step:1.2.1
Eliminate the denominator of $9$ where possible.
Step:1.2.1.1
Remove the common factor of $9$ to obtain $y = \frac{-17x}{9} + \frac{72}{9}$ and maintain $-72 + 17x = -9y$.
Step:1.2.1.2
Simplify the expression for $y$ to $y = \frac{-17x}{9} + \frac{72}{9}$ and keep $-72 + 17x = -9y$.
Step:1.3
Simplify the constant term on the right side.
Step:1.3.1
Reduce the fraction $\frac{72}{9}$.
Step:1.3.1.1
Place the negative sign in front of the fraction for $x$ and simplify the constant to get $y = -\frac{17x}{9} + 8$ and $-72 + 17x = -9y$.
Step:2
Substitute the expression for $y$ into the second equation.
Step:2.1
Replace $y$ in $-72 + 17x = -9y$ with the expression $-\frac{17x}{9} + 8$ to get $-72 + 17x = -9(-\frac{17x}{9} + 8)$.
Step:2.2
Expand and simplify the equation.
Step:2.2.1
Distribute the $-9$ across $-\frac{17x}{9} + 8$.
Step:2.2.1.1
Use the distributive property to get $-72 + 17x = 9(-\frac{17x}{9}) - 9 \cdot 8$.
Step:2.2.1.2
Cancel out the $9$s and simplify to get $-72 + 17x = -17x - 72$.
Step:2.2.1.3
Perform the multiplication to find that $-72 + 17x = 17x - 72$.
Step:3
Solve for $x$ in the simplified equation.
Step:3.1
Attempt to isolate $x$ by moving $x$ terms to one side.
Step:3.1.1
Subtract $17x$ from both sides to get $-72 = -72$.
Step:3.1.2
Notice that there are no $x$ terms left, so the equation simplifies to $-72 = -72$.
Step:4
Recognize that the equation $-72 = -72$ is an identity, meaning it is always true regardless of the value of $x$.
Step:5
Since the equation is always true, the system of equations has infinitely many solutions. The solution is the line $y = -\frac{17x}{9} + 8$.
Knowledge Notes:
System of Linear Equations: A system of linear equations consists of two or more linear equations with the same variables. The solution to the system is the set of values that satisfy all equations simultaneously.
Simplifying Equations: Simplifying an equation involves reducing fractions, combining like terms, and performing arithmetic operations to rewrite the equation in a simpler form.
Substitution Method: This method involves solving one of the equations for one variable in terms of the others and then substituting this expression into the other equations. This can help to find the solution for the variables.
Identity: An identity in algebra is an equation that is always true, no matter what values are substituted for the variables involved. If you arrive at an identity when solving a system of equations, it means that the system has infinitely many solutions.
Infinite Solutions: If a system of equations simplifies to an identity, it means that every point on one line is also on the other line, hence the lines are coincident, and there are infinitely many solutions to the system.
LaTeX Formatting: LaTeX is a typesetting system commonly used for mathematical and scientific documents. It is used here to format mathematical expressions for clarity and precision.
