Solve the system of equations.
x=6y−3
2x+7y=13 (1 point)
First, we will solve the first equation for x:
x = 6y - 3
Next, substitute this value of x into the second equation:
2(6y - 3) + 7y = 13
12y - 6 + 7y = 13
19y - 6 = 13
19y = 19
y = 1
Now, substitute this value of y back into the first equation to solve for x:
x = 6(1) - 3
x = 6 - 3
x = 3
Therefore, the solution to the system of equations is x = 3 and y = 1.
To solve the system of equations, we can use the method of substitution. Let's rearrange the first equation, x = 6y - 3, to solve for x.
x = 6y - 3 (Equation 1)
Now, substitute this expression for x in the second equation, 2x + 7y = 13.
2(6y - 3) + 7y = 13
Simplify the equation:
12y - 6 + 7y = 13
Combine like terms:
19y - 6 = 13
Add 6 to both sides of the equation:
19y = 19
Divide both sides by 19:
y = 1
Now that we have found the value of y, substitute it back into the first equation to solve for x:
x = 6(1) - 3
x = 6 - 3
x = 3
Therefore, the solution to the system of equations is x = 3 and y = 1.
To solve the system of equations, we can use the method of substitution or the method of elimination. I will explain both methods, and you can choose which one you prefer.
Method 1: Substitution
Step 1: Solve one equation for one variable in terms of the other variable.
From the first equation, we have x = 6y - 3.
Step 2: Substitute the expression for that variable into the other equation.
Substituting x in the second equation, we get 2(6y - 3) + 7y = 13.
Step 3: Simplify and solve the resulting equation.
Expanding and simplifying the equation, we get 12y - 6 + 7y = 13.
Combining like terms, we have 19y - 6 = 13.
Add 6 to both sides to isolate the term with y, resulting in 19y = 19.
Divide both sides by 19 to solve for y, giving y = 1.
Step 4: Find the value of the other variable.
Now that we know the value of y, we can substitute it back into the first equation from the system.
Substituting y = 1 into x = 6y - 3, we get x = 6(1) - 3, which simplifies to x = 6 - 3, resulting in x = 3.
So the solution to the system of equations is x = 3 and y = 1.
Method 2: Elimination
Step 1: Multiply one or both equations by constants so that the coefficients of one variable are the same, but with opposite signs.
In this case, we don’t need to perform any multiplication since the coefficients of y in the two equations are already opposites: 1 and 7.
Step 2: Add or subtract the equations to eliminate one variable.
Subtracting the first equation from the second equation, we get (2x + 7y) - (x) = 13 - (6y - 3).
Simplifying the equation by combining like terms, we have x + 7y - 6y = 13 + 3.
This simplifies to x + y = 16.
Step 3: Solve the resulting equation.
Now we can solve the simplified equation x + y = 16. Let's solve this equation for x in terms of y.
Rearranging the equation, we have x = 16 - y.
Step 4: Substitute the value of x back into one of the original equations and solve for the remaining variable.
Substituting x = 16 - y into the first equation x = 6y - 3, we get 16 - y = 6y - 3.
Adding y to both sides and subtracting 3 from both sides, we have 16 - 3 = 6y + y.
Simplifying further, we get 13 = 7y.
Dividing both sides by 7, we find y = 13/7, which simplifies to y = 1.
Step 5: Find the value of the other variable.
Substituting the value of y into the simplified equation x = 16 - y, we get x = 16 - 1, which simplifies to x = 15.
So, the solution to the system of equations is x = 15 and y = 1.
Both methods should give the same solution.