SAT · Math · Algebra
Sistemas de dos ecuaciones lineales: resolverlos y saber cuándo tienen una, ninguna o infinitas soluciones.
$x+y=10$ y $x-y=2$. Sumando: $2x=12$ → $x=6$ y $y=4$.
Intenta responder antes de abrir la solución. Las preguntas están en el idioma del examen.
$$\begin{cases} x+y=15 \\ x-y=3 \end{cases}$$
The solution to the given system of equations is $(x, y)$. What is the value of $x$?
Respuesta numérica (escribes el número).
Respuesta: 9
Add the two equations to eliminate $y$: $(x+y)+(x-y)=15+3$, so $2x=18$ and $x=9$.
Check: $y=15-9=6$, and $9-6=3$.
$x+y=14$
$x-y=4$
If $(x, y)$ is the solution to the given system of equations, what is the value of $xy$?
Respuesta correcta: B. $45$
Add the equations: $2x=18$, so $x=9$. Then $y=14-9=5$, and $xy=9\cdot5=45$.
18 is $2x$, not $xy$. 50 and 56 come from pairing the wrong values (for example, $10\cdot5$ or $14\cdot4$).
$3x+5y=41$
$5x+3y=23$
If $(x, y)$ is the solution to the given system of equations, what is the value of $y-x$?
Respuesta correcta: C. $9$
Subtract the first equation from the second: $(5x+3y)-(3x+5y)=23-41$, so $2x-2y=-18$ and $x-y=-9$. Therefore $y-x=9$.
(Solving fully: adding the equations gives $8x+8y=64$, so $x+y=8$; together with $y-x=9$, this gives $y=8.5$ and $x=-0.5$.)
$-9$ is the value of $x-y$, not $y-x$. 8 is the value of $x+y$. 18 is the value of $2(y-x)$.