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PSAT Math : How to find the solution for a system of equations

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Example Questions

Example Question #2 : Creating Equations With Whole Numbers

2x-y=2 $\displaystyle 2x-y=2$

x+y=4 $\displaystyle x+y=4$

What is the solution of $\displaystyle x$ for the two systems of equations?

Possible Answers:

x=3 $\displaystyle x=3$

x=9 $\displaystyle x=9$

x=2 $\displaystyle x=2$

x=0 $\displaystyle x=0$

x=1 $\displaystyle x=1$

Correct answer:

x=2 $\displaystyle x=2$

Explanation:

We first add both systems of equations.

Left Hand Side:

$\displaystyle (2x-y)+(x+y)=3x$

Right Hand Side:

2+4=6 $\displaystyle 2+4=6$

Our resulting equation is:

3x=6 $\displaystyle 3x=6$

We divide both sides by 3.

Left Hand Side:

$\frac{3x}{3}=x$ $\displaystyle \frac{3x}{3}=x$

Right Hand Side:

$\frac{6}{3}=2$ $\displaystyle \frac{6}{3}=2$

Our resulting equation is:

x=2 $\displaystyle x=2$

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Example Question #51 : How To Find The Solution For A System Of Equations

4x+y=8 $\displaystyle 4x+y=8$

x+4y=17 $\displaystyle x+4y=17$

What is the solution of $\displaystyle y$ for the two systems?

Possible Answers:

y=6 $\displaystyle y=6$

y=3 $\displaystyle y=3$

y=1 $\displaystyle y=1$

y=2 $\displaystyle y=2$

y=4 $\displaystyle y=4$

Correct answer:

y=4 $\displaystyle y=4$

Explanation:

We first multiply the second equation by 4.

So our resulting equation is:

$x\cdot4+4y\cdot4=17\cdot4$ $\displaystyle x\cdot4+4y\cdot4=17\cdot4$

$\displaystyle 4x+16y=68$

Then we subtract the first equation from the second new equation.

Left Hand Side:

$\displaystyle (4x+y)-(4x+16y)=-15y$

Right Hand Side:

$\displaystyle 6-68=-60$

Resulting Equation:

$\displaystyle -15y=-60$

We divide both sides by -15

Left Hand Side:

$\frac{-15y}{-15}=y$ $\displaystyle \frac{-15y}{-15}=y$

Right Hand Side:

$\frac{-60}{-15}=4$ $\displaystyle \frac{-60}{-15}=4$

Our result is:

y=4 $\displaystyle y=4$

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Example Question #11 : How To Find The Solution For A System Of Equations

Find the solutions for the following set of equations:

$\displaystyle 13x+2y=11$

$\displaystyle -5x-y=-1$

Possible Answers:

x=2 $\displaystyle x=2$

y=-7.5 $\displaystyle y=-7.5$

x=3 $\displaystyle x=3$

y=-14 $\displaystyle y=-14$

x=1 $\displaystyle x=1$

y=-1 $\displaystyle y=-1$

x=1 $\displaystyle x=1$

y=-4 $\displaystyle y=-4$

$x=\frac{9}{13}$ $\displaystyle x=\frac{9}{13}$

y=1 $\displaystyle y=1$

Correct answer:

x=3 $\displaystyle x=3$

y=-14 $\displaystyle y=-14$

Explanation:

If we multiply both sides of our bottom equation by $\displaystyle 2$, we get $\displaystyle -10x-2y=-2$. We can now add our two equations, and eliminate $\displaystyle y$, leaving only one variable. When we add the equations, we get 3x=9 $\displaystyle 3x=9$. Therefore, x=3 $\displaystyle x=3$. Finally, we go back to either of our equations, and plug in x=3 $\displaystyle x=3$ so we can solve for $\displaystyle y$.

$\displaystyle 13(3)+2y=11$

$\displaystyle 39+2y=11$

2y=-28 $\displaystyle 2y=-28$

y=-14 $\displaystyle y=-14$

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Example Question #11 : How To Find The Solution For A System Of Equations

Give the solution to the system of equations below.

$\displaystyle 4x-3y=11$

2x+y=13 $\displaystyle 2x+y=13$

Possible Answers:

(6, 1) $\displaystyle (6, 1)$

(2, -1) $\displaystyle (2, -1)$

(4, -3) $\displaystyle (4, -3)$

No solution

(5, 3) $\displaystyle (5, 3)$

Correct answer:

(5, 3) $\displaystyle (5, 3)$

Explanation:

$\displaystyle 4x-3y=11$

2x+y=13 $\displaystyle 2x+y=13$

Solve the second equation for $\displaystyle y$, allowing us to solve using the substitution method.

2x+y=13 $\displaystyle 2x+y=13$

y=13-2x $\displaystyle y=13-2x$

Substitute for $\displaystyle y$ in the first equation, and solve for $\displaystyle x$.

$\displaystyle 4x-3(13-2x)=11$

$\displaystyle 4x-39+6x=11$

$\displaystyle 10x-39=11$

10x=50 $\displaystyle 10x=50$

x=5 $\displaystyle x=5$

Now, substitute for $\displaystyle x$ in either equation; we will choose the second. This allows us to solve for $\displaystyle y$.

2x+y=13 $\displaystyle 2x+y=13$

$2\cdot 5+y=13$ $\displaystyle 2\cdot 5+y=13$

10+y=13 $\displaystyle 10+y=13$

y=3 $\displaystyle y=3$

Now we can write the solution in the notation (x,y) $\displaystyle (x,y)$, or (5,3) $\displaystyle (5,3)$.

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PSAT Math : How to find the solution for a system of equations

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #2 : Creating Equations With Whole Numbers

Example Question #51 : How To Find The Solution For A System Of Equations

Example Question #11 : How To Find The Solution For A System Of Equations

Example Question #11 : How To Find The Solution For A System Of Equations

All PSAT Math Resources

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