Bài 4: Giải hệ phương trình bằng phương pháp cộng đại số
Hướng dẫn giải Bài 24 (Trang 19 SGK Toán Đại số 9, Tập 2)
<p>Giải các hệ phương trình:</p>
<p>a, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mn>2</mn><mo>(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>)</mo><mo>+</mo><mn>3</mn><mo>(</mo><mi>x</mi><mo>-</mo><mi>y</mi><mo>)</mo><mo>=</mo><mn>4</mn></mtd></mtr><mtr><mtd><mo>(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>)</mo><mo>+</mo><mn>2</mn><mo>(</mo><mi>x</mi><mo>-</mo><mi>y</mi><mo>)</mo><mo>=</mo><mn>5</mn></mtd></mtr></mtable></mfenced></math>; b, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mn>2</mn><mo>(</mo><mi>x</mi><mo>-</mo><mn>2</mn><mo>)</mo><mo>+</mo><mn>3</mn><mo>(</mo><mn>1</mn><mo>+</mo><mi>y</mi><mo>)</mo><mo>=</mo><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>3</mn><mo>(</mo><mi>x</mi><mo>-</mo><mn>2</mn><mo>)</mo><mo>-</mo><mn>2</mn><mo>(</mo><mn>1</mn><mo>+</mo><mi>y</mi><mo>)</mo><mo>=</mo><mo>-</mo><mn>3</mn></mtd></mtr></mtable></mfenced></math></p>
<p>Giải:</p>
<p>a, Cách 1: Đặt x+y=u, x-y=v, ta có hệ phương trình (ẩn u,v):</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mn>2</mn><mi>u</mi><mo>+</mo><mn>3</mn><mi>v</mi><mo>=</mo><mn>4</mn></mtd></mtr><mtr><mtd><mi>u</mi><mo>+</mo><mn>2</mn><mi>v</mi><mo>=</mo><mn>5</mn></mtd></mtr></mtable></mfenced><mspace linebreak="newline"/><mi>n</mi><mi>ê</mi><mi>n</mi><mo> </mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mn>2</mn><mi>u</mi><mo>+</mo><mn>3</mn><mi>v</mi><mo>=</mo><mn>4</mn></mtd></mtr><mtr><mtd><mi>u</mi><mo>+</mo><mn>2</mn><mi>v</mi><mo>=</mo><mn>5</mn></mtd></mtr></mtable></mfenced><mo>⇔</mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mn>2</mn><mi>u</mi><mo>+</mo><mn>3</mn><mi>v</mi><mo>=</mo><mn>4</mn></mtd></mtr><mtr><mtd><mn>2</mn><mi>u</mi><mo>+</mo><mn>4</mn><mi>v</mi><mo>=</mo><mn>10</mn></mtd></mtr></mtable></mfenced><mo>⇔</mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mn>2</mn><mi>u</mi><mo>+</mo><mn>3</mn><mi>v</mi><mo>=</mo><mn>4</mn></mtd></mtr><mtr><mtd><mo>-</mo><mi>v</mi><mo>=</mo><mo>-</mo><mn>6</mn></mtd></mtr></mtable></mfenced><mo>⇔</mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mn>2</mn><mi>u</mi><mo>+</mo><mn>3</mn><mi>v</mi><mo>=</mo><mn>4</mn></mtd></mtr><mtr><mtd><mi>v</mi><mo>=</mo><mn>6</mn></mtd></mtr></mtable></mfenced><mo>⇔</mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mn>2</mn><mi>u</mi><mo>=</mo><mn>4</mn><mo>-</mo><mn>3</mn><mo>.</mo><mn>6</mn></mtd></mtr><mtr><mtd><mi>v</mi><mo>=</mo><mn>6</mn></mtd></mtr></mtable></mfenced><mo>⇔</mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mi>u</mi><mo>=</mo><mo>-</mo><mn>7</mn></mtd></mtr><mtr><mtd><mi>v</mi><mo>=</mo><mn>6</mn></mtd></mtr></mtable></mfenced><mspace linebreak="newline"/><mi>S</mi><mi>u</mi><mi>y</mi><mo> </mo><mi>r</mi><mi>a</mi><mo> </mo><mi>h</mi><mi>ệ</mi><mo> </mo><mi>đ</mi><mi>ã</mi><mo> </mo><mi>c</mi><mi>h</mi><mi>o</mi><mo> </mo><mi>t</mi><mi>ư</mi><mi>ơ</mi><mi>n</mi><mi>g</mi><mo> </mo><mi>đ</mi><mi>ư</mi><mi>ơ</mi><mi>n</mi><mi>g</mi><mo> </mo><mi>v</mi><mi>ớ</mi><mi>i</mi><mo>:</mo><mspace linebreak="newline"/><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mo>-</mo><mn>7</mn></mtd></mtr><mtr><mtd><mi>x</mi><mo>-</mo><mi>y</mi><mo>=</mo><mn>6</mn></mtd></mtr></mtable></mfenced><mo>⇔</mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mn>2</mn><mi>x</mi><mo>=</mo><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mi>x</mi><mo>-</mo><mi>y</mi><mo>=</mo><mn>6</mn></mtd></mtr></mtable></mfenced><mo>⇔</mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mi>x</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mn>1</mn></mrow><mn>2</mn></mfrac></mtd></mtr><mtr><mtd><mi>y</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mn>13</mn></mrow><mn>2</mn></mfrac></mtd></mtr></mtable></mfenced></math></p>
<p>Cách 2: Thu gọn vế trái của hai phương trình ta được hệ tương đương:</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mfenced open="{" close=""><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>+</mo><mn>3</mn><mi>x</mi><mo>-</mo><mn>3</mn><mi>y</mi></mrow></mfenced><mo>=</mo><mn>4</mn></mtd></mtr><mtr><mtd><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>2</mn><mi>y</mi><mo>=</mo><mn>5</mn></mtd></mtr></mtable></mfenced><mo>⇔</mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mn>5</mn><mi>x</mi><mo>-</mo><mi>y</mi><mo>=</mo><mn>4</mn></mtd></mtr><mtr><mtd><mn>3</mn><mi>x</mi><mo>-</mo><mi>y</mi><mo>=</mo><mn>5</mn></mtd></mtr></mtable></mfenced><mo>⇔</mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mn>2</mn><mi>x</mi><mo>=</mo><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>3</mn><mi>x</mi><mo>-</mo><mi>y</mi><mo>=</mo><mn>5</mn></mtd></mtr></mtable></mfenced><mo>⇔</mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mi>x</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mn>1</mn></mrow><mn>2</mn></mfrac></mtd></mtr><mtr><mtd><mi>y</mi><mo>=</mo><mn>3</mn><mi>x</mi><mo>-</mo><mn>5</mn></mtd></mtr></mtable></mfenced><mo>⇔</mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mi>x</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mn>1</mn></mrow><mn>2</mn></mfrac></mtd></mtr><mtr><mtd><mi>y</mi><mo>=</mo><mn>3</mn><mo>(</mo><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>)</mo><mo>-</mo><mn>5</mn></mtd></mtr></mtable></mfenced><mo>⇔</mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mi>x</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mn>1</mn></mrow><mn>2</mn></mfrac></mtd></mtr><mtr><mtd><mi>y</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mn>13</mn></mrow><mn>2</mn></mfrac></mtd></mtr></mtable></mfenced></math></p>
<p>b,</p>
<p>Cách 1: Đặt x-2=u, 1+y=v ta có hệ phương trình ẩn u,v:</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mn>2</mn><mi>u</mi><mo>+</mo><mn>3</mn><mi>v</mi><mo>=</mo><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>3</mn><mi>u</mi><mo>-</mo><mn>2</mn><mi>v</mi><mo>=</mo><mo>-</mo><mn>3</mn></mtd></mtr></mtable></mfenced><mspace linebreak="newline"/><mi>n</mi><mi>ê</mi><mi>n</mi><mo> </mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mn>2</mn><mi>u</mi><mo>+</mo><mn>3</mn><mi>v</mi><mo>=</mo><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>3</mn><mi>u</mi><mo>-</mo><mn>2</mn><mi>v</mi><mo>=</mo><mo>-</mo><mn>3</mn></mtd></mtr></mtable></mfenced><mo>⇔</mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mn>6</mn><mi>u</mi><mo>+</mo><mn>9</mn><mi>v</mi><mo>=</mo><mo>-</mo><mn>6</mn></mtd></mtr><mtr><mtd><mn>6</mn><mi>u</mi><mo>-</mo><mn>4</mn><mi>v</mi><mo>=</mo><mo>-</mo><mn>6</mn></mtd></mtr></mtable></mfenced><mo>⇔</mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mn>6</mn><mi>u</mi><mo>+</mo><mn>9</mn><mi>v</mi><mo>=</mo><mo>-</mo><mn>6</mn></mtd></mtr><mtr><mtd><mn>13</mn><mi>v</mi><mo>=</mo><mn>0</mn></mtd></mtr></mtable></mfenced><mo>⇔</mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mn>6</mn><mi>u</mi><mo>=</mo><mo>-</mo><mn>6</mn><mo>+</mo><mn>9</mn><mi>v</mi></mtd></mtr><mtr><mtd><mi>v</mi><mo>=</mo><mn>0</mn></mtd></mtr></mtable></mfenced><mo>⇔</mo><mfenced open="{" close=""><mrow><mtable columnalign="left"><mtr><mtd><mn>6</mn><mi>u</mi><mo>=</mo><mo>-</mo><mn>6</mn></mtd></mtr><mtr><mtd><mi>v</mi><mo>=</mo><mn>0</mn></mtd></mtr></mtable><mo>⇔</mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mi>u</mi><mo>=</mo><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mi>v</mi><mo>=</mo><mn>0</mn></mtd></mtr></mtable></mfenced></mrow></mfenced><mspace linebreak="newline"/><mi>S</mi><mi>u</mi><mi>y</mi><mo> </mo><mi>r</mi><mi>a</mi><mo> </mo><mi>h</mi><mi>ệ</mi><mo> </mo><mi>p</mi><mi>h</mi><mi>ư</mi><mi>ơ</mi><mi>n</mi><mi>g</mi><mo> </mo><mi>t</mi><mi>r</mi><mi>ì</mi><mi>n</mi><mi>h</mi><mo> </mo><mi>t</mi><mi>ư</mi><mi>ơ</mi><mi>n</mi><mi>g</mi><mo> </mo><mi>đ</mi><mi>ư</mi><mi>ơ</mi><mi>n</mi><mi>g</mi><mo> </mo><mi>v</mi><mi>ớ</mi><mi>i</mi><mo>:</mo><mspace linebreak="newline"/><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mi>x</mi><mo>-</mo><mn>2</mn><mo>=</mo><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn><mo>+</mo><mi>y</mi><mo>=</mo><mn>0</mn></mtd></mtr></mtable></mfenced><mo>⇔</mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mi>x</mi><mo>=</mo><mn>1</mn></mtd></mtr><mtr><mtd><mi>y</mi><mo>=</mo><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced></math></p>
<p>Cách 2: Thu gọn vế trái của hai phương trình:</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mn>2</mn><mo>(</mo><mi>x</mi><mo>-</mo><mn>2</mn><mo>)</mo><mo>+</mo><mn>3</mn><mo>(</mo><mn>1</mn><mo>+</mo><mi>y</mi><mo>)</mo><mo>=</mo><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>3</mn><mo>(</mo><mi>x</mi><mo>-</mo><mn>2</mn><mo>)</mo><mo>-</mo><mn>2</mn><mo>(</mo><mn>1</mn><mo>+</mo><mi>y</mi><mo>)</mo><mo>=</mo><mo>-</mo><mn>3</mn></mtd></mtr></mtable></mfenced><mo>⇔</mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mn>2</mn><mi>x</mi><mo>-</mo><mn>4</mn><mo>+</mo><mn>3</mn><mo>+</mo><mn>3</mn><mi>y</mi><mo>=</mo><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>3</mn><mi>x</mi><mo>-</mo><mn>6</mn><mo>-</mo><mn>2</mn><mo>-</mo><mn>2</mn><mi>y</mi><mo>=</mo><mo>-</mo><mn>3</mn></mtd></mtr></mtable></mfenced><mo>⇔</mo><mfenced open="{" close=""><mrow><mtable columnalign="left"><mtr><mtd><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>=</mo><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>3</mn><mi>x</mi><mo>-</mo><mn>2</mn><mi>y</mi><mo>=</mo><mn>5</mn></mtd></mtr></mtable><mo>⇔</mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mn>6</mn><mi>x</mi><mo>+</mo><mn>9</mn><mi>y</mi><mo>=</mo><mo>-</mo><mn>3</mn></mtd></mtr><mtr><mtd><mn>6</mn><mi>x</mi><mo>-</mo><mn>4</mn><mi>y</mi><mo>=</mo><mn>10</mn></mtd></mtr></mtable></mfenced></mrow></mfenced><mo>⇔</mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mn>6</mn><mi>x</mi><mo>+</mo><mn>9</mn><mi>y</mi><mo>=</mo><mo>-</mo><mn>3</mn></mtd></mtr><mtr><mtd><mn>13</mn><mi>y</mi><mo>=</mo><mo>-</mo><mn>13</mn></mtd></mtr></mtable></mfenced><mo>⇔</mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mn>6</mn><mi>x</mi><mo>=</mo><mo>-</mo><mn>3</mn><mo>-</mo><mn>9</mn><mi>y</mi></mtd></mtr><mtr><mtd><mi>y</mi><mo>=</mo><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced><mo>⇔</mo><mfenced open="{" close=""><mrow><mtable columnalign="left"><mtr><mtd><mn>6</mn><mi>x</mi><mo>=</mo><mn>6</mn></mtd></mtr><mtr><mtd><mi>y</mi><mo>=</mo><mo>-</mo><mn>1</mn></mtd></mtr></mtable><mo>⇔</mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mi>x</mi><mo>=</mo><mn>1</mn></mtd></mtr><mtr><mtd><mi>y</mi><mo>=</mo><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mrow></mfenced><mspace linebreak="newline"/></math></p>
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