{"id":676932,"date":"2023-09-02T19:43:17","date_gmt":"2023-09-02T14:13:17","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/question\/the-pair-of-equations-3x4yk-9x12y6-has-infinitely-many-solutions-if\/"},"modified":"2025-06-26T17:37:47","modified_gmt":"2025-06-26T12:07:47","slug":"the-pair-of-equations-3x4yk-9x12y6-has-infinitely-many-solutions-if","status":"publish","type":"questions","link":"https:\/\/infinitylearn.com\/surge\/question\/mathematics\/the-pair-of-equations-3x4yk9x12y6-has-infinitely-many\/","title":{"rendered":"The pair of equations 3x+4y=k,\u00a09x+12y=6 has infinitely many solutions if"},"content":{"rendered":"","protected":false},"author":1,"template":"","meta":{"_yoast_wpseo_focuskw":"","_yoast_wpseo_title":"","_yoast_wpseo_metadesc":"The pair of equations 3x+4y=k,\u00a09x+12y=6 has infinitely many solutions if","custom_permalink":"question\/mathematics\/the-pair-of-equations-3x4yk9x12y6-has-infinitely-many\/"},"categories":[],"acf":{"question":"

The pair of equations <\/span>3<\/mn>x<\/mi>+<\/mo>4<\/mn>y<\/mi>=<\/mo>k<\/mi>,<\/mo> <\/mi>9<\/mn>x<\/mi>+<\/mo>12<\/mn>y<\/mi>=<\/mo>6<\/mn><\/math> has infinitely many solutions if <\/span><\/p>

<\/p>

<\/p><\/spanstyle=\"font-family:'timesnewroman';-aw-import:ignore\"><\/pstyle=\"margin-top:0pt;margin-bottom:0pt;line-height:normal;font-size:12pt\"><\/spanstyle=\"font-family:'timesnewroman';-aw-import:ignore\"><\/pstyle=\"margin-top:0pt;margin-bottom:0pt;line-height:normal;font-size:12pt\">","options":[{"option":"k<\/mi>=<\/mo>2<\/mn><\/math> <\/span>","correct":true},{"option":"k<\/mi>=<\/mo>6<\/mn><\/math> <\/span>","correct":false},{"option":"k <\/mi>\u2260<\/mo> <\/mi>6<\/mn><\/math> <\/span>","correct":false},{"option":"k<\/mi>=<\/mo>3<\/mn><\/math> <\/span> <\/span>","correct":false}],"solution":"The given equations are,<\/span>
3<\/mn>x<\/mi>+<\/mo>4<\/mn>y<\/mi>-<\/mo>k<\/mi>=<\/mo>0<\/mn><\/math> <\/span>9<\/mn>x<\/mi>+<\/mo>12<\/mn>y<\/mi>-<\/mo>6<\/mn>=<\/mo>0<\/mn><\/math> <\/span>Here, <\/span>a<\/mi><\/mrow>1<\/mn><\/mrow><\/msub>=<\/mo>3<\/mn>,<\/mo>b<\/mi><\/mrow>1<\/mn><\/mrow><\/msub>=<\/mo>4<\/mn>,<\/mo>c<\/mi><\/mrow>1<\/mn><\/mrow><\/msub>=<\/mo>-<\/mo>k<\/mi><\/math>,<\/span>
And, <\/span>a<\/mi><\/mrow>2<\/mn><\/mrow><\/msub>=<\/mo>9<\/mn>,<\/mo>b<\/mi><\/mrow>2<\/mn><\/mrow><\/msub>=<\/mo>12<\/mn>,<\/mo>c<\/mi><\/mrow>2<\/mn><\/mrow><\/msub>=<\/mo>-<\/mo>6<\/mn><\/math>.<\/span>
For the system to have infinite solutions condition is<\/span>
 <\/span>a<\/mi><\/mrow>1<\/mn><\/mrow><\/msub><\/mrow>a<\/mi><\/mrow>2<\/mn><\/mrow><\/msub><\/mrow><\/mfrac>=<\/mo>b<\/mi><\/mrow>1<\/mn><\/mrow><\/msub><\/mrow>b<\/mi><\/mrow>2<\/mn><\/mrow><\/msub><\/mrow><\/mfrac>=<\/mo>c<\/mi><\/mrow>1<\/mn><\/mrow><\/msub><\/mrow>c<\/mi><\/mrow>2<\/mn><\/mrow><\/msub><\/mrow><\/mfrac><\/math> <\/span> <\/span>Here, <\/span>3<\/mn><\/mrow>9<\/mn><\/mrow><\/mfrac>=<\/mo>4<\/mn><\/mrow>12<\/mn><\/mrow><\/mfrac>=<\/mo>-<\/mo>k<\/mi><\/mrow>-<\/mo>6<\/mn><\/mrow><\/mfrac><\/math>
\u21d2 <\/span>4<\/mn><\/mrow>12<\/mn><\/mrow><\/mfrac>=<\/mo>-<\/mo>k<\/mi><\/mrow>-<\/mo>6<\/mn><\/mrow><\/mfrac><\/math> <\/span>\u21d2<\/mo>1<\/mn><\/mrow>3<\/mn><\/mrow><\/mfrac>=<\/mo>-<\/mo>k<\/mi><\/mrow>-<\/mo>6<\/mn><\/mrow><\/mfrac><\/math> <\/span>\u21d2 <\/span>3<\/mn>k<\/mi>=<\/mo>6<\/mn><\/math> <\/span>\u21d2 <\/span>k<\/mi>=<\/mo>2<\/mn><\/math> <\/span>Hence, the correct option is 1.<\/span>
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