Solve the system of equations: y=6x 4x+y=7
Understand the Problem
The question asks to solve a system of two equations with two variables, x and y. We can use substitution to solve for x and y.
Answer
$x = \frac{16}{7}$, $y = \frac{11}{7}$
Answer for screen readers
$x = \frac{16}{7}$, $y = \frac{11}{7}$
Steps to Solve
- Solve the first equation for x
We have $x + 3y = 7$. Subtract $3y$ from both sides to isolate $x$: $$x = 7 - 3y$$
- Substitute the expression for x into the second equation
Substitute $x = 7 - 3y$ into the second equation $2x - y = 3$: $$2(7 - 3y) - y = 3$$
- Simplify and solve for y
Expand and simplify the equation: $$14 - 6y - y = 3$$ $$14 - 7y = 3$$ Subtract 14 from both sides: $$-7y = 3 - 14$$ $$-7y = -11$$ Divide by -7: $$y = \frac{-11}{-7} = \frac{11}{7}$$
- Substitute the value of y back into the equation for x
Substitute $y = \frac{11}{7}$ into $x = 7 - 3y$: $$x = 7 - 3\left(\frac{11}{7}\right)$$ $$x = 7 - \frac{33}{7}$$ $$x = \frac{49}{7} - \frac{33}{7}$$ $$x = \frac{16}{7}$$
$x = \frac{16}{7}$, $y = \frac{11}{7}$
More Information
The solution to the system of equations is $x = \frac{16}{7}$ and $y = \frac{11}{7}$. This means that the point $(\frac{16}{7}, \frac{11}{7})$ is the intersection of the two lines represented by the given equations.
Tips
A common mistake is to make an error when distributing or combining like terms in the algebraic manipulations. Another mistake is to incorrectly substitute the value of one variable back into the equation to solve for the other variable. Careful attention to detail and double-checking each step can help avoid these errors.
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