Here are the step-by-step solutions for each system of equations using the substitution method.
1)
System:
$$y = -7$$
$$4x + 3y = -10$$
Step 1: The first equation already tells us that $y = -7$. We can substitute this value directly into the second equation.
$$4x + 3(-7) = -10$$
Step 2: Multiply $3$ by $-7$.
$$4x - 21 = -10$$
Step 3: Add $21$ to both sides to isolate the term with $x$.
$$4x = -10 + 21$$
$$4x = 11$$
Step 4: Divide by $4$ to solve for $x$.
$$x = \frac{11}{4}$$ or $2.75$
The solution is $x = \frac{11}{4}, y = -7$.
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2)
System:
$$-2x - 9y = -7$$
$$y = 8x - 11$$
Step 1: Substitute $(8x - 11)$ for $y$ in the first equation.
$$-2x - 9(8x - 11) = -7$$
Step 2: Distribute the $-9$ into the parentheses. Be careful with signs: $-9 \times 8x = -72x$ and $-9 \times -11 = +99$.
$$-2x - 72x + 99 = -7$$
Step 3: Combine like terms ($-2x$ and $-72x$).
$$-74x + 99 = -7$$
Step 4: Subtract $99$ from both sides.
$$-74x = -7 - 99$$
$$-74x = -106$$
Step 5: Divide by $-74$.
$$x = \frac{-106}{-74}$$
Simplify the fraction by dividing top and bottom by $2$:
$$x = \frac{53}{37}$$
Step 6: Find $y$ by plugging $x$ back into the second equation ($y = 8x - 11$).
$$y = 8\left(\frac{53}{37}\right) - 11$$
$$y = \frac{424}{37} - 11$$
To subtract, convert $11$ to a fraction with a denominator of $37$ ($11 \times 37 = 407$).
$$y = \frac{424}{37} - \frac{407}{37}$$
$$y = \frac{17}{37}$$
The solution is $x = \frac{53}{37}, y = \frac{17}{37}$.
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3)
System:
$$y = x + 5$$
$$2x - 3y = -2$$
Step 1: Substitute $(x + 5)$ for $y$ in the second equation.
$$2x - 3(x + 5) = -2$$
Step 2: Distribute the $-3$.
$$2x - 3x - 15 = -2$$
Step 3: Combine like terms ($2x - 3x = -x$).
$$-x - 15 = -2$$
Step 4: Add $15$ to both sides.
$$-x = -2 + 15$$
$$-x = 13$$
$$x = -13$$
Step 5: Find $y$ using the first equation ($y = x + 5$).
$$y = -13 + 5$$
$$y = -8$$
The solution is $x = -13, y = -8$.
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4)
System:
$$5x + 9y = 2$$
$$y = -3x + 5$$
Step 1: Substitute $(-3x + 5)$ for $y$ in the first equation.
$$5x + 9(-3x + 5) = 2$$
Step 2: Distribute the $9$.
$$5x - 27x + 45 = 2$$
Step 3: Combine like terms ($5x - 27x = -22x$).
$$-22x + 45 = 2$$
Step 4: Subtract $45$ from both sides.
$$-22x = 2 - 45$$
$$-22x = -43$$
Step 5: Divide by $-22$.
$$x = \frac{-43}{-22}$$
$$x = \frac{43}{22}$$
Step 6: Find $y$ using the second equation ($y = -3x + 5$).
$$y = -3\left(\frac{43}{22}\right) + 5$$
$$y = \frac{-129}{22} + 5$$
Convert $5$ to a fraction with denominator $22$ ($5 \times 22 = 110$).
$$y = \frac{-129}{22} + \frac{110}{22}$$
$$y = \frac{-19}{22}$$
The solution is $x = \frac{43}{22}, y = -\frac{19}{22}$.
Final Answer:
1) $x = \frac{11}{4}, y = -7$
2) $x = \frac{53}{37}, y = \frac{17}{37}$
3) $x = -13, y = -8$
4) $x = \frac{43}{22}, y = -\frac{19}{22}$
Parent Tip: Review the logic above to help your child master the concept of system of equations by substitution worksheet.
