Solve the System of Equations y-10=3x 2y=6x+20
The problem consists of finding the values of the variables x and y that satisfy two linear equations simultaneously. The first equation is y - 10 = 3x, and the second equation is 2y = 6x + 20. The task is to determine the specific x and y that make both equations true when substituting these values into the respective equations. This process is commonly referred to as solving a system of linear equations, and it typically involves methods such as substitution, elimination, or graphing to find the point of intersection, which represents the solution to the system.
$y - 10 = 3 x$$2 y = 6 x + 20$
Answer
Solution:
Step 1:
Increment both sides of the first equation by $10$ to isolate $y$.
$$y = 3x + 10$$ $$2y = 6x + 20$$
Step 2:
Substitute the expression for $y$ from the first equation into the second equation.
Step 2.1:
In the second equation $2y = 6x + 20$, replace $y$ with $3x + 10$.
$$2(3x + 10) = 6x + 20$$ $$y = 3x + 10$$
Step 2.2:
Expand the expression on the left side of the equation.
Step 2.2.1:
Use the distributive property to expand $2(3x + 10)$.
Step 2.2.1.1:
Distribute $2$ across the terms within the parentheses.
$$2(3x) + 2 \cdot 10 = 6x + 20$$ $$y = 3x + 10$$
Step 2.2.1.2:
Perform the multiplication.
Step 2.2.1.2.1:
Multiply $3$ by $2$.
$$6x + 2 \cdot 10 = 6x + 20$$ $$y = 3x + 10$$
Step 2.2.1.2.2:
Multiply $2$ by $10$.
$$6x + 20 = 6x + 20$$ $$y = 3x + 10$$
Step 3:
Eliminate any redundant equations from the system that are identities.
$$y = 3x + 10$$
Step 4:
The remaining equation represents the solution to the system.
Knowledge Notes:
In solving a system of linear equations, there are several methods that can be used, such as substitution, elimination, and graphing. The substitution method involves solving one of the equations for one variable and then substituting that expression into the other equation(s). This method is particularly useful when one of the equations is already solved for one variable or can be easily manipulated to do so.
The steps involved in the substitution method include:
Solve one of the equations for one variable if it is not already done.
Substitute the expression obtained into the other equation(s).
Simplify the resulting equation(s) and solve for the variable(s).
Substitute back to find the value of the other variable(s).
In this problem, the first equation was already solved for $y$, making it easy to substitute into the second equation. The distributive property was used to expand the expression $2(3x + 10)$, which is a fundamental algebraic property that allows us to multiply a single term by each term inside a parenthesis. After simplifying, we found that the second equation did not provide new information, as it became an identity (an equation that is true for all values of the variables involved). Therefore, the solution to the system is given by the remaining equation, $y = 3x + 10$.
