Solve the system of equations: y=6x, 4x+y=7
Understand the Problem
The question requires solving a system of two equations with two variables (x and y). We can use substitution to solve for x and y.
Answer
$x = 2$, $y = 1$
Answer for screen readers
$x = 2$ and $y = 1$
Steps to Solve
-
Solve the first equation for $x$ We have the equation $x + 3y = 5$. We can isolate $x$ by subtracting $3y$ from both sides. $$x = 5 - 3y$$
-
Substitute the expression for $x$ into the second equation The second equation is $2x - y = 3$. Substitute $x = 5 - 3y$ into this equation. $$2(5 - 3y) - y = 3$$
-
Simplify and solve for $y$ Distribute the $2$ and combine like terms. $$10 - 6y - y = 3$$ $$10 - 7y = 3$$ Subtract $10$ from both sides: $$-7y = -7$$ Divide by $-7$: $$y = 1$$
-
Substitute the value of $y$ back into the expression for $x$ We found that $y = 1$, and we have $x = 5 - 3y$. Substitute $y = 1$ into this equation. $$x = 5 - 3(1)$$ $$x = 5 - 3$$ $$x = 2$$
-
State the solution The solution is $x = 2$ and $y = 1$.
$x = 2$ and $y = 1$
More Information
We can check the solution by substituting the values of $x$ and $y$ into the original equations: $x + 3y = 2 + 3(1) = 2 + 3 = 5$ $2x - y = 2(2) - 1 = 4 - 1 = 3$ Both equations are satisfied.
Tips
A common mistake is to incorrectly distribute when substituting the expression for one variable into the other equation. For example, in step 3, forgetting to distribute the 2 to both terms in $(5-3y)$ would lead to an incorrect value of $y$. Also, errors in sign manipulation during algebraic simplification are frequent sources of mistakes.
AI-generated content may contain errors. Please verify critical information
