Question

systems of equations study guide
solve the problems below. be sure to show your thinking.

1. mateo plans to solve the system below using elimination. which is a reasonable first step mateo could take?

-x + 6y = 9
3x + 2y = 13
a. multiply the 2nd equation by 3
b. multiply the 2nd equation by -3
c. multiply the 1st equation by -3
d. any of the above

1. in baseball, a single is worth 1 base and a double is worth 2 bases. suppose a player has 9 hits that total 15 bases. how many doubles did the player hit?
2. what can you conclude from the graph of the equations shown below?

a. the solution to the equations y = -2x + 18 and y = \\(\frac{4}{3}\\)x + 4 is (8, 10).
b. the solution to the equations y = -x + 18 and y = \\(\frac{4}{3}\\)x + 4 is (10, 8).
c. the solution to the equations y = -x + 18 and y = \\(\frac{3}{4}\\)x + 4 is (10, 8).
d. the solution to the equations y = -x + 18 and y = \\(\frac{3}{4}\\)x + 4 is (8, 10).
graph of two lines intersecting on a coordinate plane

1. solve the system of equations by graphing.

y = \\(\frac{2}{3}\\)x - 2
y = -4
coordinate plane grid

1. solve the system of equations by graphing.

y = 2x
x + y = -6
coordinate plane grid

Explanation:

Problem 1

Step1: Analyze elimination for x terms

First equation: $-x + 6y = 9$; second: $3x + 2y = 13$. Multiply first equation by 3: $-3x + 18y = 27$, which cancels $3x$ in second equation.

Step2: Analyze elimination for y terms

Multiply second equation by -3: $-9x -6y = -39$, which cancels $6y$ in first equation. Multiply second by 3: $9x +6y=39$, which also works to cancel $6y$ via subtraction.

Step3: Evaluate all options

All options A, B, C are valid first steps.

Problem 2

Step1: Define variables

Let $s$ = singles, $d$ = doubles.

Step2: Set up system of equations

Total hits: $s + d = 9$; total bases: $s + 2d = 15$.

Step3: Eliminate s via subtraction

Subtract first equation from second: $(s+2d)-(s+d)=15-9$
$\implies d=6$

Problem 3

Step1: Identify line equations

Downward line: y-intercept 18, slope $\frac{2-18}{16-0}=-1$, so $y=-x+18$. Upward line: y-intercept 4, slope $\frac{10-4}{8-0}=\frac{3}{4}$, so $y=\frac{3}{4}x+4$.

Step2: Verify intersection point

Intersection is $(8,10)$: plug into $y=-x+18$: $10=-8+18=10$; plug into $y=\frac{3}{4}x+4$: $10=\frac{3}{4}(8)+4=6+4=10$.

Problem 4

Step1: Find intersection of lines

Set $\frac{2}{3}x -2 = -4$

Step2: Solve for x

$\frac{2}{3}x = -4 + 2 = -2$
$\implies x = -2 \times \frac{3}{2} = -3$
Intersection point: $(-3, -4)$

Problem 5

Step1: Rewrite second equation

$x + y = -6 \implies y = -x -6$

Step2: Find intersection with $y=2x$

Set $2x = -x -6$

Step3: Solve for x

$2x + x = -6 \implies 3x = -6 \implies x=-2$

Step4: Find y value

$y=2(-2)=-4$
Intersection point: $(-2, -4)$

Answer:

1. D. Any of the above
2. 6
3. D. The solution to the equations $y = -x + 18$ and $y = \frac{3}{4}x + 4$ is $(8, 10)$.
4. $(-3, -4)$
5. $(-2, -4)$