Solve the following simultaneous equations: \(y = x^2 + 3x + 7\) \(y - x = 10\)
Understand the Problem
The question requires solving a system of two equations for (x) and (y). The equations are (y = x^2 + 3x + 7) and (y - x = 10). We need to find the values of (x) and (y) that satisfy both equations simultaneously.
Answer
$x = -3, y = 7$ and $x = 1, y = 11$
Answer for screen readers
The solutions to the simultaneous equations are: $x = -3, y = 7$ and $x = 1, y = 11$.
Steps to Solve
- Express $y$ in terms of $x$ using the second equation
From the equation $y - x = 10$, we have $y = x + 10$.
- Substitute the expression for $y$ into the first equation
Substituting $y = x + 10$ into $y = x^2 + 3x + 7$ gives us $x + 10 = x^2 + 3x + 7$.
- Rearrange the equation into a quadratic equation
Rearranging the equation $x + 10 = x^2 + 3x + 7$ to standard quadratic form, we get $0 = x^2 + 2x - 3$.
- Solve the quadratic equation
We can factor the quadratic equation $x^2 + 2x - 3 = 0$ as $(x + 3)(x - 1) = 0$. This gives us two possible solutions for $x$: $x = -3$ or $x = 1$.
- Find the corresponding values for $y$
For $x = -3$, we have $y = x + 10 = -3 + 10 = 7$. For $x = 1$, we have $y = x + 10 = 1 + 10 = 11$.
- Write the solutions for $x$ and $y$
The solutions are $(x, y) = (-3, 7)$ and $(x, y) = (1, 11)$.
The solutions to the simultaneous equations are: $x = -3, y = 7$ and $x = 1, y = 11$.
More Information
Simultaneous equations are a set of equations containing multiple variables where we aim to find values for the variables that satisfy all equations at the same time.
Tips
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