Solving systems of equations by elimination or by substitution ... - Free Printable

Here are the step-by-step solutions for each system of equations using the substitution method.

1)

System:
$$m + n = 3$$
$$m - n = 5$$

Step 1: Solve the first equation for $m$.
$$m = 3 - n$$

Step 2: Substitute $(3 - n)$ for $m$ in the second equation.
$$(3 - n) - n = 5$$
$$3 - 2n = 5$$

Step 3: Solve for $n$.
$$-2n = 5 - 3$$
$$-2n = 2$$
$$n = -1$$

Step 4: Substitute $n = -1$ back into the equation from Step 1 to find $m$.
$$m = 3 - (-1)$$
$$m = 3 + 1$$
$$m = 4$$

Check:
$4 + (-1) = 3$ (Correct)
$4 - (-1) = 5$ (Correct)

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2)

System:
$$2x + 3y = 32$$
$$3x + y = 10$$

Step 1: Solve the second equation for $y$ (it's the easiest one to isolate).
$$y = 10 - 3x$$

Step 2: Substitute $(10 - 3x)$ for $y$ in the first equation.
$$2x + 3(10 - 3x) = 32$$
$$2x + 30 - 9x = 32$$

Step 3: Solve for $x$.
$$-7x + 30 = 32$$
$$-7x = 2$$
$$x = -\frac{2}{7}$$

Step 4: Substitute $x = -\frac{2}{7}$ back into the equation from Step 1 to find $y$.
$$y = 10 - 3(-\frac{2}{7})$$
$$y = 10 + \frac{6}{7}$$
To add these, convert 10 to a fraction with a denominator of 7: $10 = \frac{70}{7}$.
$$y = \frac{70}{7} + \frac{6}{7} = \frac{76}{7}$$

Check:
$2(-\frac{2}{7}) + 3(\frac{76}{7}) = -\frac{4}{7} + \frac{228}{7} = \frac{224}{7} = 32$ (Correct)
$3(-\frac{2}{7}) + \frac{76}{7} = -\frac{6}{7} + \frac{76}{7} = \frac{70}{7} = 10$ (Correct)

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3)

System:
$$12m + 10n = 35$$
$$-m + 2n = 17$$

Step 1: Solve the second equation for $m$.
$$-m = 17 - 2n$$
Multiply by -1:
$$m = -17 + 2n$$ (or $2n - 17$)

Step 2: Substitute $(2n - 17)$ for $m$ in the first equation.
$$12(2n - 17) + 10n = 35$$
$$24n - 204 + 10n = 35$$

Step 3: Solve for $n$.
$$34n - 204 = 35$$
$$34n = 239$$
$$n = \frac{239}{34}$$

Step 4: Substitute $n = \frac{239}{34}$ back into the equation from Step 1 to find $m$.
$$m = 2(\frac{239}{34}) - 17$$
$$m = \frac{478}{34} - 17$$
Simplify $\frac{478}{34}$ to $\frac{239}{17}$. Convert 17 to $\frac{289}{17}$.
$$m = \frac{239}{17} - \frac{289}{17}$$
$$m = -\frac{50}{17}$$

*(Note: These numbers are complex fractions. Let's double check the arithmetic. $12(2n-17)+10n=35 \rightarrow 24n-204+10n=35 \rightarrow 34n=239$. The math is correct based on the numbers provided.)*

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4)

System:
$$4x - y = 1$$
$$2x + 3y = 11$$

Step 1: Solve the first equation for $y$.
$$-y = 1 - 4x$$
$$y = 4x - 1$$

Step 2: Substitute $(4x - 1)$ for $y$ in the second equation.
$$2x + 3(4x - 1) = 11$$
$$2x + 12x - 3 = 11$$

Step 3: Solve for $x$.
$$14x - 3 = 11$$
$$14x = 14$$
$$x = 1$$

Step 4: Substitute $x = 1$ back into the equation from Step 1 to find $y$.
$$y = 4(1) - 1$$
$$y = 3$$

Check:
$4(1) - 3 = 1$ (Correct)
$2(1) + 3(3) = 2 + 9 = 11$ (Correct)

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5)

System:
$$7a - 2b = 14$$
$$5a + 4b = -9$$

Step 1: Solve the first equation for $2b$ or isolate $b$. Let's isolate $2b$ first to make substitution easier, or just solve for $b$.
$$7a - 14 = 2b$$
$$b = \frac{7a - 14}{2}$$
Alternatively, solve for $7a$: $7a = 14 + 2b \rightarrow a = 2 + \frac{2}{7}b$. This looks messy. Let's stick to substituting for $b$ or use elimination logic within substitution.
Let's solve the first equation for $2b$: $2b = 7a - 14$.
Notice the second equation has $4b$, which is $2(2b)$.
Substitute $(7a - 14)$ for $2b$ in the second equation?
$5a + 2(2b) = -9$
$5a + 2(7a - 14) = -9$

Step 2: Solve for $a$.
$$5a + 14a - 28 = -9$$
$$19a - 28 = -9$$
$$19a = 19$$
$$a = 1$$

Step 3: Substitute $a = 1$ back into the expression for $2b$.
$$2b = 7(1) - 14$$
$$2b = -7$$
$$b = -3.5$$ (or $-\frac{7}{2}$)

Check:
$7(1) - 2(-3.5) = 7 + 7 = 14$ (Correct)
$5(1) + 4(-3.5) = 5 - 14 = -9$ (Correct)

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6)

System:
$$-4x + 2y = 3$$
$$2x + 3y = -2$$

Step 1: Solve the second equation for $2x$.
$$2x = -2 - 3y$$
Notice the first equation has $-4x$, which is $-2(2x)$.
Substitute $(-2 - 3y)$ for $2x$ in the first equation:
$$-2(2x) + 2y = 3$$
$$-2(-2 - 3y) + 2y = 3$$

Step 2: Solve for $y$.
$$4 + 6y + 2y = 3$$
$$4 + 8y = 3$$
$$8y = -1$$
$$y = -\frac{1}{8}$$

Step 3: Substitute $y = -\frac{1}{8}$ back into the equation for $2x$.
$$2x = -2 - 3(-\frac{1}{8})$$
$$2x = -2 + \frac{3}{8}$$
Convert -2 to $-\frac{16}{8}$.
$$2x = -\frac{16}{8} + \frac{3}{8}$$
$$2x = -\frac{13}{8}$$
Divide by 2:
$$x = -\frac{13}{16}$$

Check:
$-4(-\frac{13}{16}) + 2(-\frac{1}{8}) = \frac{52}{16} - \frac{2}{8} = \frac{13}{4} - \frac{1}{4} = \frac{12}{4} = 3$ (Correct)
$2(-\frac{13}{16}) + 3(-\frac{1}{8}) = -\frac{26}{16} - \frac{3}{8} = -\frac{13}{8} - \frac{3}{8} = -\frac{16}{8} = -2$ (Correct)

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Final Answer:
1) $m = 4, n = -1$
2) $x = -\frac{2}{7}, y = \frac{76}{7}$
3) $m = -\frac{50}{17}, n = \frac{239}{34}$
4) $x = 1, y = 3$
5) $a = 1, b = -3.5$ (or $-\frac{7}{2}$)
6) $x = -\frac{13}{16}, y = -\frac{1}{8}$