Question

algebra i homework
fluency

1. solve each of the following system of equations by substitution.

(a) $y = x + 8$
$y = 4x - 1$
(b) $y = -3x + 5$
$2x + y = 6$
(c) $4x + 3y = 37$
$y = x - 4$
(d) $x - 5y = -49$
$y = -2x + 1$

1. given the system shown below do the following:

$y = \frac{1}{2}x - 2$
$y = -3x + 5$
(a) solve this system graphically using the grid shown.
(b) solve this system by substitution. show your work.
(grid: x from -10 to 10, y from -10 to 10, with grid lines)

Explanation:

---

Problem 1(a)

Step1: Substitute $y=x+8$ into $y=4x-1$

$x+8=4x-1$

Step2: Isolate $x$ terms

$8+1=4x-x$
$9=3x$

Step3: Solve for $x$

$x=\frac{9}{3}=3$

Step4: Solve for $y$ using $y=x+8$

$y=3+8=11$

---

Problem 1(b)

Step1: Substitute $y=-3x+5$ into $2x+y=6$

$2x+(-3x+5)=6$

Step2: Simplify left side

$2x-3x+5=6$
$-x+5=6$

Step3: Solve for $x$

$-x=6-5$
$x=-1$

Step4: Solve for $y$ using $y=-3x+5$

$y=-3(-1)+5=3+5=8$

---

Problem 1(c)

Step1: Substitute $y=x-4$ into $4x+3y=37$

$4x+3(x-4)=37$

Step2: Expand and simplify

$4x+3x-12=37$
$7x-12=37$

Step3: Solve for $x$

$7x=37+12$
$7x=49$
$x=7$

Step4: Solve for $y$ using $y=x-4$

$y=7-4=3$

---

Problem 1(d)

Step1: Substitute $y=-2x+1$ into $x-5y=-49$

$x-5(-2x+1)=-49$

Step2: Expand and simplify

$x+10x-5=-49$
$11x-5=-49$

Step3: Solve for $x$

$11x=-49+5$
$11x=-44$
$x=-4$

Step4: Solve for $y$ using $y=-2x+1$

$y=-2(-4)+1=8+1=9$

---

Problem 2(b)

Step1: Substitute $y=\frac{1}{2}x-2$ into $y=-3x+5$

$\frac{1}{2}x-2=-3x+5$

Step2: Eliminate fraction, multiply by 2

$x-4=-6x+10$

Step3: Isolate $x$ terms

$x+6x=10+4$
$7x=14$

Step4: Solve for $x$

$x=2$

Step5: Solve for $y$ using $y=-3x+5$

$y=-3(2)+5=-6+5=-1$

---

Problem 2(a)

Step1: Find intercepts for $y=\frac{1}{2}x-2$

• x-intercept: $0=\frac{1}{2}x-2 \implies x=4$ (point $(4,0)$)
• y-intercept: $y=-2$ (point $(0,-2)$)

Step2: Find intercepts for $y=-3x+5$

• x-intercept: $0=-3x+5 \implies x=\frac{5}{3}\approx1.67$ (point $(\frac{5}{3},0)$)
• y-intercept: $y=5$ (point $(0,5)$)

Step3: Plot lines, find intersection

Lines intersect at $(2,-1)$

Answer:

1. (a) $x=3$, $y=11$
2. (b) $x=-1$, $y=8$
3. (c) $x=7$, $y=3$
4. (d) $x=-4$, $y=9$
5. (a) The solution is the intersection point $(2,-1)$
6. (b) $x=2$, $y=-1$