5x - 7y = -19; -7x + 4y = -14
Understand the Problem
The question provides a system of equations and asks for a solution, likely through methods such as substitution or elimination.
Answer
The solution is \( x = 6 \) and \( y = 7 \).
Answer for screen readers
The solution to the system of equations is ( x = 6 ) and ( y = 7 ).
Steps to Solve
- Rearranging One Equation for Substitution
Select the first equation and rearrange it to solve for $y$:
Starting with the equation:
$$ 5x - 7y = -19 $$
Isolate $y$:
$$ -7y = -19 - 5x $$
Divide by -7:
$$ y = \frac{5x + 19}{7} $$
- Substituting into the Second Equation
Now, substitute the expression for $y$ into the second equation:
The second equation is:
$$ -7x + 4y = -14 $$
Substituting $y$ from the first equation gives:
$$ -7x + 4\left(\frac{5x + 19}{7}\right) = -14 $$
- Clearing the Fraction
To eliminate the fraction, multiply the entire equation by 7:
$$ -49x + 4(5x + 19) = -98 $$
Distributing the 4:
$$ -49x + 20x + 76 = -98 $$
- Combining Like Terms
Combine the $x$ terms:
$$ -29x + 76 = -98 $$
- Isolating x
Subtract 76 from both sides:
$$ -29x = -174 $$
Divide by -29:
$$ x = \frac{174}{29} = 6 $$
- Substituting back to find y
Now substitute $x = 6$ back into the equation we derived for $y$:
$$ y = \frac{5(6) + 19}{7} $$
Calculate:
$$ y = \frac{30 + 19}{7} = \frac{49}{7} = 7 $$
The solution to the system of equations is ( x = 6 ) and ( y = 7 ).
More Information
This is a linear system of equations, and the solution shows the point at which both lines represented by the equations intersect. This point can be confirmed by substituting ( x ) and ( y ) back into the original equations.
Tips
- Failing to distribute correctly when substituting ( y ) into the second equation.
- Forgetting to simplify fractions before solving for ( x ) or ( y ).
- Confusing the signs when isolating variables.
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