Solve the system of equations: y=6x and 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
$x = 2$, $y = 1$
Answer for screen readers
$x = 2$ and $y = 1$ or $(2, 1)$
Steps to Solve
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Solve the first equation for x We start with the first equation: $x + 3y = 5$. Subtract $3y$ from both sides to isolate $x$. $x = 5 - 3y$
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Substitute the expression for x into the second equation The second equation is $2x - y = 3$. Replace $x$ with the expression we found in step 1, $5 - 3y$. This gives us: $2(5 - 3y) - y = 3$
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Simplify and solve for y Distribute the 2: $10 - 6y - y = 3$ Combine like terms: $10 - 7y = 3$ Subtract 10 from both sides: $-7y = -7$ Divide by -7: $y = 1$
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Substitute the value of y back into the equation for x We found that $x = 5 - 3y$. Substitute $y = 1$ into this equation: $x = 5 - 3(1)$ $x = 5 - 3$ $x = 2$
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State the solution The solution is $x = 2$ and $y = 1$. We can write this as an ordered pair $(2, 1)$.
$x = 2$ and $y = 1$ or $(2, 1)$
More Information
We can check the solution by plugging the values of $x$ and $y$ into both original equations:
Equation 1: $2 + 3(1) = 2 + 3 = 5$, which is correct. Equation 2: $2(2) - 1 = 4 - 1 = 3$, which is also correct.
Tips
A common mistake is to make an error when distributing or combining like terms. For example, forgetting to distribute the 2 in the equation $2(5 - 3y) - y = 3$ would lead to an incorrect solution. Another common mistake is solving for the wrong variable or substituting the value back into the wrong equation. Careful attention to detail can avoid these mistakes.
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